1:"$Sreact.fragment"
2:I[10156,[],""]
3:I[6746,[],""]
4:I[76837,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","104","static/chunks/104-e8a10a728f64230d.js","3374","static/chunks/3374-cffcb39c106cc278.js","6344","static/chunks/6344-f0148458c0499269.js","2043","static/chunks/2043-cb8fa44c52a8f8b9.js","9559","static/chunks/app/(frontend)/layout-42f64306bd0733c5.js"],"default"]
5:I[32043,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","104","static/chunks/104-e8a10a728f64230d.js","3374","static/chunks/3374-cffcb39c106cc278.js","6344","static/chunks/6344-f0148458c0499269.js","2043","static/chunks/2043-cb8fa44c52a8f8b9.js","9559","static/chunks/app/(frontend)/layout-42f64306bd0733c5.js"],"default"]
6:I[67821,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],""]
1a:I[14600,[],"default",1]
:HL["/_next/static/media/849c382dee069680-s.p.woff2","font",{"crossOrigin":"","type":"font/woff2"}]
:HL["/_next/static/media/e4af272ccee01ff0-s.p.woff2","font",{"crossOrigin":"","type":"font/woff2"}]
:HL["/_next/static/css/60d9b09af57ce3c6.css","style"]
:HL["/_next/static/css/9c59f2e83b7f05d1.css","style"]
0:{"P":null,"c":["","calculators","statistics","binomial-cdf-calculator.rsc"],"q":"","i":false,"f":[[["",{"children":["(frontend)",{"children":["calculators",{"children":[["category","statistics","d",null],{"children":[["slug","binomial-cdf-calculator","d",null],{"children":["__PAGE__",{}]}]}]}]},"$undefined","$undefined",16]}],[["$","$1","c",{"children":[null,["$","$L2",null,{"parallelRouterKey":"children","error":"$undefined","errorStyles":"$undefined","errorScripts":"$undefined","template":["$","$L3",null,{}],"templateStyles":"$undefined","templateScripts":"$undefined","notFound":[[["$","title",null,{"children":"404: This page could not be found."}],["$","div",null,{"style":{"fontFamily":"system-ui,\"Segoe UI\",Roboto,Helvetica,Arial,sans-serif,\"Apple Color Emoji\",\"Segoe UI Emoji\"","height":"100vh","textAlign":"center","display":"flex","flexDirection":"column","alignItems":"center","justifyContent":"center"},"children":["$","div",null,{"children":[["$","style",null,{"dangerouslySetInnerHTML":{"__html":"body{color:#000;background:#fff;margin:0}.next-error-h1{border-right:1px solid rgba(0,0,0,.3)}@media (prefers-color-scheme:dark){body{color:#fff;background:#000}.next-error-h1{border-right:1px solid rgba(255,255,255,.3)}}"}}],["$","h1",null,{"className":"next-error-h1","style":{"display":"inline-block","margin":"0 20px 0 0","padding":"0 23px 0 0","fontSize":24,"fontWeight":500,"verticalAlign":"top","lineHeight":"49px"},"children":404}],["$","div",null,{"style":{"display":"inline-block"},"children":["$","h2",null,{"style":{"fontSize":14,"fontWeight":400,"lineHeight":"49px","margin":0},"children":"This page could not be found."}]}]]}]}]],[]],"forbidden":"$undefined","unauthorized":"$undefined"}]]}],{"children":[["$","$1","c",{"children":[[["$","link","0",{"rel":"stylesheet","href":"/_next/static/css/60d9b09af57ce3c6.css","precedence":"next","crossOrigin":"$undefined","nonce":"$undefined"}],["$","link","1",{"rel":"stylesheet","href":"/_next/static/css/9c59f2e83b7f05d1.css","precedence":"next","crossOrigin":"$undefined","nonce":"$undefined"}]],["$","html",null,{"lang":"en","data-scroll-behavior":"smooth","children":[["$","body",null,{"className":"bg-white antialiased dark:bg-mist-950 __variable_f367f3 __variable_008b00","suppressHydrationWarning":true,"children":[["$","div",null,{"className":"flex min-h-screen flex-col justify-between","children":[["$","$L4",null,{}],["$","$L5",null,{"children":["$","$L2",null,{"parallelRouterKey":"children","error":"$undefined","errorStyles":"$undefined","errorScripts":"$undefined","template":["$","$L3",null,{}],"templateStyles":"$undefined","templateScripts":"$undefined","notFound":[["$","main",null,{"className":"isolate overflow-clip","children":["$","section",null,{"className":"py-16","children":["$","div",null,{"className":"mx-auto w-full max-w-2xl px-6 md:max-w-3xl lg:max-w-7xl lg:px-10 flex flex-col items-center gap-6","children":[["$","h1",null,{"className":"font-display text-5xl/12 tracking-tight text-balance sm:text-[5rem]/20 text-mist-950 dark:text-white max-w-5xl text-center","children":"Page not found"}],["$","div",null,{"className":"text-lg/8 text-mist-700 dark:text-mist-400 max-w-xl text-center","children":["$","p",null,{"children":"Sorry, the page you were looking for could not be found."}]}],["$","$L6",null,{"href":"/","className":"inline-flex items-center gap-2 text-sm/7 font-medium text-mist-950 dark:text-white","children":["Go back home ",["$","svg",null,{"width":13,"height":7,"viewBox":"0 0 13 7","fill":"none","strokeWidth":1,"role":"image","className":"inline-block","children":[["$","path",null,{"d":"M12.5049 3.49512L0.504883 3.49512","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round"}],["$","path",null,{"d":"M9.5 6.5L12.5 3.5L9.5 0.5","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round"}]]}]]}]]}]}]}],[]],"forbidden":"$undefined","unauthorized":"$undefined"}]}],["$","footer",null,{"children":["$","div",null,{"className":"bg-mist-950/2.5 py-16 text-mist-950 dark:bg-white/5 dark:text-white","children":["$","div",null,{"className":"mx-auto w-full max-w-2xl px-6 md:max-w-3xl lg:max-w-7xl lg:px-10 flex flex-col gap-12","children":[["$","div",null,{"className":"flex flex-col items-start justify-between gap-6 sm:flex-row","children":[["$","div",null,{"className":"flex flex-col items-start gap-2","children":[["$","$L6",null,{"href":"/","children":["$","svg",null,{"className":"h-5 w-auto","xmlns":"http://www.w3.org/2000/svg","width":742,"height":68,"viewBox":"0 0 742 68","fill":"none","children":[["$","path",null,{"fill":"#161616","d":"M129.5 41.4c0-13 10.5-23.4 23.4-23.4 12.9 0 23.3 10.4 23.3 23.4v22.7h-11.8v-4.8c-1.4 3-6.3 5.5-11.5 5.5-12.9 0-23.4-10.4-23.4-23.4Zm35 0c0-6.7-4.8-11.9-11.5-11.9s-11.7 5.2-11.7 11.9 5 11.9 11.7 11.9c6.7 0 11.5-5.2 11.5-11.9ZM181.2 41.4c0-13 10.4-23.4 23.4-23.4 13 0 19.8 7.2 22.5 16.9h-12.5c-2-3.3-5.5-5.4-9.9-5.4-6.7 0-11.7 5.2-11.7 11.9s5 11.9 11.7 11.9c6.7 0 7.4-2 9.5-5H227c-2.9 9.6-11.8 16.5-22.4 16.5-10.6 0-23.4-10.4-23.4-23.4ZM248.3 30.3v12.4c0 7.3 2.4 10.2 8.2 10.2v11.5c-11.6 0-19.9-6.4-19.9-21.7V30.3h-6.7V18.8h6.6V1.1h11.8v17.7h8.3v11.5H248.3ZM274 18.7h-11.8V64H274V18.7ZM277.8 18.7h12.3l10.3 29.8 10.4-29.8h12.3L306.2 64h-11.8l-16.8-45.3h.2ZM345.8 18c11.2 0 23.2 8 23.7 22.4 0 2.1 0 4.1-.4 5.6h-33.6c1.1 4.5 5.1 7.3 10.3 7.3 5.2 0 6.3-1.5 7.8-2.6h14c-3.6 8.3-11.9 14.1-21.5 14.1s-23.4-10.4-23.4-23.3 10-23.5 23-23.5h.1Zm11 18.6c-1.8-4.3-6.2-7.1-10.7-7.1-4.5 0-8.9 2.8-10.7 7.1h21.4Z"}],"$L7","$L8","$L9","$La","$Lb","$Lc"]}]}],"$Ld"]}],"$Le"]}],"$Lf","$L10"]}]}]}]]}],"$L11","$L12"]}],"$L13"]}]]}],{"children":["$L14",{"children":["$L15",{"children":["$L16",{"children":["$L17",{},null,false,null]},null,false,"$@18"]},null,false,"$@18"]},null,false,"$@18"]},null,false,null]},null,false,"$@18"],"$L19",false]],"m":"$undefined","G":["$1a",[]],"S":true,"h":null,"b":"qKrY2dUw1szY2xJfsSyIe"}
1c:I[14245,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","104","static/chunks/104-e8a10a728f64230d.js","3374","static/chunks/3374-cffcb39c106cc278.js","6344","static/chunks/6344-f0148458c0499269.js","2043","static/chunks/2043-cb8fa44c52a8f8b9.js","9559","static/chunks/app/(frontend)/layout-42f64306bd0733c5.js"],"Button"]
22:I[46726,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","104","static/chunks/104-e8a10a728f64230d.js","3374","static/chunks/3374-cffcb39c106cc278.js","6344","static/chunks/6344-f0148458c0499269.js","2043","static/chunks/2043-cb8fa44c52a8f8b9.js","9559","static/chunks/app/(frontend)/layout-42f64306bd0733c5.js"],"default"]
23:I[19592,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","6344","static/chunks/6344-f0148458c0499269.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","2018","static/chunks/app/(frontend)/%5B...slug%5D/page-dcf70230194846a1.js"],""]
24:I[79066,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","6344","static/chunks/6344-f0148458c0499269.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","2018","static/chunks/app/(frontend)/%5B...slug%5D/page-dcf70230194846a1.js"],"GoogleTagManager"]
26:I[85305,[],"OutletBoundary"]
27:"$Sreact.suspense"
2a:I[85305,[],"ViewportBoundary"]
2c:I[85305,[],"MetadataBoundary"]
1b:T5a9,M373.9 41.4c0-13 10.5-23.4 23.4-23.4 12.9 0 16.9 5.3 20.6 12.8h-6.3c-3.2-4.4-8.2-7.3-14.2-7.3-10 0-17.7 7.9-17.7 17.9 0 10 7.9 17.9 17.7 17.9 9.8 0 11.1-3 14.2-7.3h6.4c-3.7 7.6-11.6 12.9-20.8 12.9-9.2 0-23.4-10.4-23.4-23.4l.1-.1ZM422.7 41.4c0-13 10.5-23.4 23.4-23.4 12.9 0 23.3 10.4 23.3 23.4v22.7h-5.8v-9.7c-3 6-9.8 10.4-17.5 10.4-12.9 0-23.4-10.4-23.4-23.4Zm41 0c0-10-7.5-17.9-17.5-17.9s-17.7 7.9-17.7 17.9c0 10 7.9 17.9 17.7 17.9 9.8 0 17.5-7.9 17.5-17.9ZM475.7 64.1V.7h5.8v63.4h-5.8ZM486.5 41.4c0-13 10.5-23.4 23.4-23.4 12.9 0 16.9 5.3 20.6 12.8h-6.3c-3.2-4.4-8.2-7.3-14.2-7.3-10 0-17.7 7.9-17.7 17.9 0 10 7.9 17.9 17.7 17.9 9.8 0 11.1-3 14.2-7.3h6.4c-3.7 7.6-11.6 12.9-20.8 12.9-9.2 0-23.4-10.4-23.4-23.4l.1-.1ZM575.6 18.7v23.8c0 15.2-8.3 22.2-19.9 22.2-11.6 0-19.9-7-19.9-22.2V18.7h5.8v23.8c0 11.3 5.7 16.7 14.1 16.7 8.4 0 14.1-5.3 14.1-16.7V18.7h5.8ZM581.9 64.1V.7h5.8v63.4h-5.8ZM592.7 41.4c0-13 10.5-23.4 23.4-23.4 12.9 0 23.3 10.4 23.3 23.4v22.7h-5.8v-9.7c-3 6-9.8 10.4-17.5 10.4-12.9 0-23.4-10.4-23.4-23.4Zm41 0c0-10-7.5-17.9-17.5-17.9s-17.7 7.9-17.7 17.9c0 10 7.9 17.9 17.7 17.9 9.8 0 17.5-7.9 17.5-17.9ZM652.5 24.3v18.2c0 10.1 4.4 15.1 11.4 16.2v5.6c-10.2-1-17.2-7.8-17.2-21.7V24.2h-6.8v-5.5h6.8V4.4h5.8v14.3h11.4v5.5h-11.4v.1ZM712 41.4c0 13-10.4 23.4-23.4 23.4-13 0-23.4-10.4-23.4-23.4 0-13 10.5-23.4 23.4-23.4 12.9 0 23.4 10.4 23.4 23.4Zm-41 0c0 10 7.9 17.9 17.7 17.9 9.8 0 17.5-7.9 17.5-17.9 0-10-7.5-17.9-17.5-17.9S671 31.4 671 41.4Z7:["$","path",null,{"fill":"#262626","d":"$1b"}]
8:["$","path",null,{"fill":"#09C269","d":"M274 1.1h-11.8v11.8H274V1.1Z"}]
9:["$","path",null,{"fill":"#161616","d":"M248.2 1.1v17.6L236.4 1.1h11.8Z"}]
a:["$","path",null,{"fill":"#09C269","d":"M65.2 47.1c-4.1-5.4-2.8-11.8-4.4-18-1.1-4.2-3.1-8.2-5.8-11.6-.7-.9-1.5-1.8-2.3-2.6-5.6-5.6-13.3-9-21.8-9C13.1 5.9-1.1 21 .2 39c1.3 18 13 26.9 27.9 28.2 9.6.8 18.4-2.8 24.6-9l-.4-.4c-3-3.8-5.1-8-6.3-12.6-3.3 5.7-9.7 9.4-16.8 8.7-8.1-.8-14.7-7.3-15.6-15.3-.9-8 7-19.4 17.3-19.4S43 22.7 46 27.9c2.6 4.6 2 9.6 3 14.6.9 4.8 2.9 9.3 6 13.2 5.6 7.1 14.3 11.6 24 11.6h30.8V55.2H82.1c-6.3 0-12.4-2.7-16.4-7.5-.2-.2-.4-.5-.6-.7l.1.1Z"}]
b:["$","path",null,{"fill":"#09C269","d":"M64 27.9c3-5.2 8.6-8.8 15.1-8.8 9.6 0 17.4 7.8 17.4 17.4h13.4c0-17-13.8-30.8-30.8-30.8s-16.2 3.5-21.8 9.1l.4.5c3 3.8 5.1 8 6.3 12.6Z"}]
c:["$","path",null,{"fill":"#262626","d":"M717.3 40.6c0-15.5 7.9-22.6 19.1-22.6h1.4v5.7h-1.4c-7.8 0-13.1 5.4-13.1 16.9v24.2h-6V40.6Z"}]
d:["$","p",null,{"className":"text-sm text-mist-600 dark:text-mist-400","children":"Interactive calculators that convert visitors into leads."}]
e:["$","$L1c",null,{"el":"a","href":"https://app.activecalculator.com/auth/signup","appearance":"default","label":"Get started free","data-umami-event":"click-cta-footer","data-umami-event-location":"footer"}]
f:["$","nav",null,{"className":"grid grid-cols-2 gap-6 text-sm/7 md:grid-cols-4","children":[["$","div","Use Cases",{"children":[["$","h3",null,{"children":"Use Cases"}],["$","ul",null,{"role":"list","className":"mt-2 flex flex-col gap-2","children":[["$","li","E-Commerce & Retail",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/category/e-commerce","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"E-Commerce & Retail"}]}],["$","li","Finance",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/category/finance","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Finance"}]}],["$","li","Unit Conversion",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/category/unit-conversion","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Unit Conversion"}]}],["$","li","Price Quote",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/category/price-quote","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Price Quote"}]}]]}],[["$","div","Industries",{"className":"mt-6","children":[["$","h3",null,{"children":"Industries"}],["$","ul",null,{"role":"list","className":"mt-2 flex flex-col gap-2","children":[["$","li","Screen Printing",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/industries/screen-printing","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Screen Printing"}]}],["$","li","Mortgage Brokers",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/industries/mortgage-brokers","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Mortgage Brokers"}]}],["$","li","Solar Energy",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/industries/solar-energy","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Solar Energy"}]}]]}]]}]]]}],["$","div","Templates",{"children":[["$","h3",null,{"children":"Templates"}],["$","ul",null,{"role":"list","className":"mt-2 flex flex-col gap-2","children":[["$","li","Ad Spend ROI Calculator",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/ad-spend-roi-calculator","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Ad Spend ROI Calculator"}]}],["$","li","Amortization Payment Calculator Canada",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/amortization-payment-calculator-canada","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Amortization Payment Calculator Canada"}]}],["$","li","Amortization Calculator",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/amortization-calculator","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Amortization Calculator"}]}],["$","li","Balloon Loan Calculator Lead Gen",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates/balloon-loan-calculator-lead-gen","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Balloon Loan Calculator Lead Gen"}]}]]}],"$undefined"]}],["$","div","Product",{"children":[["$","h3",null,{"children":"Product"}],["$","ul",null,{"role":"list","className":"mt-2 flex flex-col gap-2","children":[["$","li","Templates",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/templates","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Templates"}]}],["$","li","Calculators",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/calculators","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Calculators"}]}],["$","li","Pricing",{"className":"text-mist-700 dark:text-mist-400","children":"$L1d"}],"$L1e","$L1f","$L20"]}],"$undefined"]}],"$L21"]}]
10:["$","div",null,{"className":"flex flex-col items-start justify-between gap-4 border-t border-mist-950/10 pt-8 text-sm/7 sm:flex-row sm:items-center dark:border-white/10","children":[["$","div",null,{"className":"flex flex-wrap items-center gap-x-4 gap-y-1 text-mist-600 dark:text-mist-500","children":[["$","span",null,{"children":["© ",2026," ActiveCalculator."]}],[["$","$L6","Privacy Policy",{"href":"/privacy","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Privacy Policy"}],["$","$L6","Terms of Service",{"href":"/terms","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Terms of Service"}]]]}],["$","div",null,{"className":"flex items-center gap-4","children":[["$","a","X",{"href":"https://twitter.com/activecalculator","target":"_blank","rel":"noopener noreferrer","aria-label":"X","data-umami-event":"click-social","data-umami-event-location":"footer","data-umami-event-target":"x","className":"text-mist-600 transition-colors *:size-5 hover:text-mist-950 dark:text-mist-500 dark:hover:text-white","children":["$","svg",null,{"width":24,"height":24,"viewBox":"0 0 24 24","fill":"currentColor","role":"image","className":"inline-block","children":["$","path",null,{"d":"M13.6833 10.6218L20.2401 3H18.6864L12.9931 9.61788L8.44583 3H3.20117L10.0775 13.0074L3.20117 21H4.75501L10.7673 14.0113L15.5695 21H20.8141L13.6833 10.6218ZM11.5551 13.0956L10.8584 12.0991L5.31488 4.16971H7.7015L12.1752 10.5689L12.8719 11.5655L18.6871 19.8835H16.3005L11.5551 13.0956Z"}]}]}],["$","a","LinkedIn",{"href":"https://linkedin.com/company/activecalculator","target":"_blank","rel":"noopener noreferrer","aria-label":"LinkedIn","data-umami-event":"click-social","data-umami-event-location":"footer","data-umami-event-target":"linkedin","className":"text-mist-600 transition-colors *:size-5 hover:text-mist-950 dark:text-mist-500 dark:hover:text-white","children":["$","svg",null,{"width":24,"height":24,"viewBox":"0 0 24 24","fill":"currentColor","role":"image","className":"inline-block","children":["$","path",null,{"d":"M20.447 20.452h-3.554v-5.569c0-1.328-.027-3.037-1.852-3.037-1.853 0-2.136 1.445-2.136 2.939v5.667H9.351V9h3.414v1.561h.046c.477-.9 1.637-1.85 3.37-1.85 3.601 0 4.267 2.37 4.267 5.455v6.286zM5.337 7.433a2.062 2.062 0 01-2.063-2.065 2.064 2.064 0 112.063 2.065zm1.782 13.019H3.555V9h3.564v11.452zM22.225 0H1.771C.792 0 0 .774 0 1.729v20.542C0 23.227.792 24 1.771 24h20.451C23.2 24 24 23.227 24 22.271V1.729C24 .774 23.2 0 22.222 0h.003z"}]}]}],["$","a","YouTube",{"href":"https://youtube.com/@activecalculator","target":"_blank","rel":"noopener noreferrer","aria-label":"YouTube","data-umami-event":"click-social","data-umami-event-location":"footer","data-umami-event-target":"youtube","className":"text-mist-600 transition-colors *:size-5 hover:text-mist-950 dark:text-mist-500 dark:hover:text-white","children":["$","svg",null,{"width":24,"height":24,"viewBox":"0 0 24 24","fill":"currentColor","role":"image","className":"inline-block","children":["$","path",null,{"fillRule":"evenodd","d":"M19.812 5.418c.861.23 1.538.907 1.768 1.768C21.998 8.746 22 12 22 12s0 3.255-.418 4.814a2.504 2.504 0 0 1-1.768 1.768c-1.56.419-7.814.419-7.814.419s-6.255 0-7.814-.419a2.505 2.505 0 0 1-1.768-1.768C2 15.255 2 12 2 12s0-3.255.417-4.814a2.507 2.507 0 0 1 1.768-1.768C5.744 5 11.998 5 11.998 5s6.255 0 7.814.418ZM15.194 12 10 15V9l5.194 3Z","clipRule":"evenodd"}]}]}]]}]]}]
11:["$","$L22",null,{}]
12:["$","$L23",null,{"defer":true,"src":"https://analytics.selvo.co/script.js","data-website-id":"20975f57-f91a-4d7f-a72e-bc1f75a733ed","strategy":"afterInteractive"}]
13:["$","$L24",null,{"gtmId":"GTM-N55KJQ5L"}]
14:["$","$1","c",{"children":[null,["$","$L2",null,{"parallelRouterKey":"children","error":"$undefined","errorStyles":"$undefined","errorScripts":"$undefined","template":["$","$L3",null,{}],"templateStyles":"$undefined","templateScripts":"$undefined","notFound":"$undefined","forbidden":"$undefined","unauthorized":"$undefined"}]]}]
15:["$","$1","c",{"children":[null,["$","$L2",null,{"parallelRouterKey":"children","error":"$undefined","errorStyles":"$undefined","errorScripts":"$undefined","template":["$","$L3",null,{}],"templateStyles":"$undefined","templateScripts":"$undefined","notFound":"$undefined","forbidden":"$undefined","unauthorized":"$undefined"}]]}]
16:["$","$1","c",{"children":[null,["$","$L2",null,{"parallelRouterKey":"children","error":"$undefined","errorStyles":"$undefined","errorScripts":"$undefined","template":["$","$L3",null,{}],"templateStyles":"$undefined","templateScripts":"$undefined","notFound":"$undefined","forbidden":"$undefined","unauthorized":"$undefined"}]]}]
17:["$","$1","c",{"children":["$L25",null,["$","$L26",null,{"children":["$","$27",null,{"name":"Next.MetadataOutlet","children":"$@28"}]}]]}]
29:[]
18:"$W29"
19:["$","$1","h",{"children":[null,["$","$L2a",null,{"children":"$L2b"}],["$","div",null,{"hidden":true,"children":["$","$L2c",null,{"children":["$","$27",null,{"name":"Next.Metadata","children":"$L2d"}]}]}],["$","meta",null,{"name":"next-size-adjust","content":""}]]}]
1d:["$","$L6",null,{"href":"/pricing","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Pricing"}]
1e:["$","li","Help",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/help","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Help"}]}]
1f:["$","li","API",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"https://docs.activecalculator.com/api-reference","target":"_blank","rel":"noopener noreferrer","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"API"}]}]
20:["$","li","Embed SDK JS",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"https://docs.activecalculator.com/sdks/embed-sdk/overview","target":"_blank","rel":"noopener noreferrer","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Embed SDK JS"}]}]
21:["$","div","Resources",{"children":[["$","h3",null,{"children":"Resources"}],["$","ul",null,{"role":"list","className":"mt-2 flex flex-col gap-2","children":[["$","li","Terms",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/terms","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Terms"}]}],["$","li","Privacy",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/privacy","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Privacy"}]}],["$","li","Book a Call",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/demo","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Book a Call"}]}],["$","li","Contact",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"/contact","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Contact"}]}],["$","li","Selvo",{"className":"text-mist-700 dark:text-mist-400","children":["$","$L6",null,{"href":"https://selvo.co","target":"_blank","rel":"noopener noreferrer","className":"transition-colors hover:text-mist-950 dark:hover:text-white","children":"Selvo"}]}]]}],"$undefined"]}]
2b:[["$","meta","0",{"charSet":"utf-8"}],["$","meta","1",{"name":"viewport","content":"width=device-width, initial-scale=1"}],["$","meta","2",{"name":"theme-color","content":"#fff"}]]
2e:I[62540,[],"IconMark"]
28:null
2d:[["$","title","0",{"children":"Binomial CDF Calculator | Cumulative Binomial Probability"}],["$","meta","1",{"name":"description","content":"Calculate cumulative binomial probabilities instantly. Enter your trials, successes, and probability to get P(X=x), P(X≤x), P(X≥x), and more."}],["$","link","2",{"rel":"canonical","href":"https://activecalculator.com/calculators/statistics/binomial-cdf-calculator"}],["$","meta","3",{"property":"og:title","content":"Binomial CDF Calculator | Cumulative Binomial Probability"}],["$","meta","4",{"property":"og:description","content":"Calculate cumulative binomial probabilities instantly. Enter your trials, successes, and probability to get P(X=x), P(X≤x), P(X≥x), and more."}],["$","meta","5",{"property":"og:url","content":"https://activecalculator.com/calculators/statistics/binomial-cdf-calculator"}],["$","meta","6",{"property":"og:site_name","content":"ActiveCalculator"}],["$","meta","7",{"property":"og:locale","content":"en_US"}],["$","meta","8",{"property":"og:image","content":"https://activecalculator.com/api/og/calculator?title=Binomial%20CDF%20Calculator"}],["$","meta","9",{"property":"og:image:width","content":"1200"}],["$","meta","10",{"property":"og:image:height","content":"630"}],["$","meta","11",{"property":"og:image:alt","content":"Binomial CDF Calculator | Cumulative Binomial Probability"}],["$","meta","12",{"property":"og:type","content":"website"}],["$","meta","13",{"name":"twitter:card","content":"summary_large_image"}],["$","meta","14",{"name":"twitter:site","content":"@activecalculator"}],["$","meta","15",{"name":"twitter:creator","content":"@activecalculator"}],["$","meta","16",{"name":"twitter:title","content":"Binomial CDF Calculator | Cumulative Binomial Probability"}],["$","meta","17",{"name":"twitter:description","content":"Calculate cumulative binomial probabilities instantly. Enter your trials, successes, and probability to get P(X=x), P(X≤x), P(X≥x), and more."}],["$","meta","18",{"name":"twitter:image","content":"https://activecalculator.com/api/og/calculator?title=Binomial%20CDF%20Calculator"}],["$","link","19",{"rel":"icon","href":"/favicon.ico?756f06c5b2066fec","type":"image/x-icon","sizes":"48x48"}],["$","$L2e","20",{}]]
2f:T10eb,{"@type":"FAQPage","mainEntity":[{"name":"What does binomcdf stand for?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"Binomcdf stands for \"binomial cumulative distribution function.\" It calculates the probability of getting x or fewer successes in n independent trials, each with the same probability of success p. If you've used a TI-84 calculator, the binomcdf function works the same way — this online calculator just makes it accessible from any device."}},{"name":"How do I calculate P(X >= x) using binomial CDF?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"You can get P(X ≥ x) from the CDF using the complement: P(X ≥ x) = 1 − P(X ≤ x − 1). But with this calculator, you don't need to do that math yourself. Just enter your values and read P(X ≥ x) directly from the results."}},{"name":"What's the difference between binomcdf and binompdf?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"Binompdf calculates the probability of exactly x successes (a single point), while binomcdf calculates the probability of x or fewer successes (a cumulative total). For example, binompdf(10, 0.5, 3) gives the probability of exactly 3 heads in 10 coin flips, while binomcdf(10, 0.5, 3) gives the probability of 0, 1, 2, or 3 heads."}},{"name":"Can I use this calculator for a range of values like P(3 <= X <= 7)?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"Yes, but it takes two steps. Calculate P(X ≤ 7) and subtract P(X ≤ 2). The difference gives you P(3 ≤ X ≤ 7). Run the calculator once with x = 7 to get the upper bound, and once with x = 2 to get the value to subtract."}},{"name":"What if my probability of success is greater than 0.5?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"The calculator works for any probability between 0 and 1. A high probability of success simply means successes are more likely than failures, which shifts the distribution to the right. The formulas and calculations work exactly the same way."}},{"name":"How many trials can this calculator handle?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"The calculator handles a wide range of trial numbers. For very large values of n (say, over 1,000), the binomial distribution closely approximates a normal distribution, and you might find a normal approximation calculator useful as a cross-check. For most practical problems, this calculator will give you accurate results."}},{"name":"Why are my results showing 0 or 1?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"If you see probabilities of 0 or very close to 0, it usually means the outcome you're asking about is extremely unlikely given your inputs. Similarly, a probability of 1 (or very close to it) means the outcome is nearly certain. Try adjusting your number of successes (x) closer to n × p (the expected value) to see more intermediate probabilities."}},{"name":"When should I use binomial CDF vs. the normal approximation?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"For small to moderate sample sizes (n < 30 or so), use the exact binomial CDF — that's what this calculator provides. When n is large and p isn't too close to 0 or 1, the normal approximation with continuity correction becomes a reasonable shortcut. As a rule of thumb, the normal approximation works well when both n × p ≥ 5 and n × (1 − p) ≥ 5."}},{"name":"What are common mistakes when using binomial CDF?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"The most common mistakes are mixing up \"fewer than\" and \"at most\" (P(X < 5) vs. P(X ≤ 5) — they differ by exactly P(X = 5)), forgetting that trials must be independent, and using the binomial model when the probability changes between trials. Also watch for off-by-one errors: \"at least 5\" means P(X ≥ 5), not P(X > 5)."}},{"name":"Is this the same as the binomcdf function on a TI-84?","@type":"Question","acceptedAnswer":{"@type":"Answer","text":"Yes, this calculator produces the same results as the TI-84's binomcdf function. On a TI-84, you'd access it through 2nd > VARS > binomcdf(n, p, x). This online version saves you from navigating calculator menus and gives you all five probability types at once, rather than just the cumulative P(X ≤ x)."}}]}25:[["$","script",null,{"dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"name\":\"Home\",\"@type\":\"ListItem\",\"item\":\"https://activecalculator.com\",\"position\":1},{\"name\":\"Calculators\",\"@type\":\"ListItem\",\"item\":\"https://activecalculator.com/calculators\",\"position\":2},{\"name\":\"Statistics & Data Analysis\",\"@type\":\"ListItem\",\"item\":\"https://activecalculator.com/calculators/statistics\",\"position\":3},{\"name\":\"Binomial CDF Calculator\",\"@type\":\"ListItem\",\"item\":\"https://activecalculator.com/calculators/statistics/binomial-cdf-calculator\",\"position\":4}]}"},"type":"application/ld+json"}],["$","script",null,{"dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Article\",\"name\":\"Binomial CDF Calculator\",\"dateModified\":\"2026-03-06T17:25:09.101Z\",\"description\":\"Calculate cumulative binomial probabilities instantly. Enter your trials, successes, and probability to get P(X=x), P(X≤x), P(X≥x), and more.\",\"author\":{\"name\":\"ActiveCalculator\",\"@type\":\"Organization\"},\"datePublished\":\"2026-03-06T17:23:13.400Z\",\"headline\":\"Binomial CDF Calculator\",\"image\":\"https://activecalculator.com/api/og/calculator?title=Binomial%20CDF%20Calculator\",\"mainEntityOfPage\":{\"@id\":\"https://activecalculator.com/calculators/statistics/binomial-cdf-calculator\",\"@type\":\"WebPage\"},\"publisher\":{\"name\":\"ActiveCalculator\",\"@type\":\"Organization\",\"logo\":{\"@type\":\"ImageObject\",\"height\":112,\"url\":\"https://activecalculator.com/logo.png\",\"width\":112}},\"keywords\":\"Binomial CDF Calculator, tool, Statistics & Data Analysis, Statistics & Data Analysis calculator\",\"url\":\"https://activecalculator.com/calculators/statistics/binomial-cdf-calculator\"}"},"type":"application/ld+json"}],["$","script",null,{"dangerouslySetInnerHTML":{"__html":"$2f"},"type":"application/ld+json"}],"$L30"]
31:I[65569,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],"JumplistProvider"]
32:I[52456,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],"RichText"]
33:I[52924,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],"ShareButtons"]
34:I[56472,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],"RenderBlocks"]
30:["$","div",null,{"className":"min-h-screen bg-white","children":["$","$L31",null,{"children":["$","div",null,{"className":"py-10 lg:py-12 container","children":[["$","div",null,{"className":"mb-6","children":["$","div",null,{"className":"flex items-center flex-wrap gap-1 text-mist-600","children":[["$","$L6",null,{"className":"typo-small font-medium hover:text-blue-600 transition-colors duration-200","href":"/calculators","children":"Calculators"}],["$","svg",null,{"className":"text-mist-600","fill":"none","height":"16","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":"2","viewBox":"0 0 24 24","width":"16","xmlns":"http://www.w3.org/2000/svg","children":["$","polyline",null,{"points":"9 18 15 12 9 6"}]}],["$","$L6",null,{"className":"typo-small font-medium hover:text-blue-600 transition-colors duration-200","href":"/calculators/statistics","children":"Statistics & Data Analysis"}]]}]}],["$","div",null,{"className":"mb-8","children":[["$","h1",null,{"className":"typo-title mb-4 text-mist-950","children":"Binomial CDF Calculator"}],["$","div",null,{"className":"text-mist-600 typo-normal max-w-2xl mb-4","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Calculate cumulative binomial probabilities instantly. Enter your probability of success, number of trials, and target successes to get P(X=x), P(X≤x), P(X≥x), and more.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}],["$","div",null,{"className":"flex mt-4","children":["$","$L33",null,{"variant":"full","title":"Binomial CDF Calculator"}]}]]}],false,["$","div",null,{"className":"grid grid-cols-1 lg:grid-cols-6 gap-8","children":[["$","div",null,{"className":"lg:col-span-4","children":[["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-8 mb-6","children":["$","$L34",null,{"blocks":[{"blockType":"blogContent","blogContentFields":{"settings":{"background":"default","paddingTop":"medium","paddingBottom":"medium"},"richText":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This binomial CDF calculator helps you find cumulative probabilities for binomial distributions quickly and accurately. Whether you're working through a statistics assignment, analyzing quality control data, or running a probability experiment, just enter your values and get results for all five common probability types — P(X = x), P(X < x), P(X ≤ x), P(X > x), and P(X ≥ x).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"No more plugging numbers into formulas by hand or scrolling through probability tables. Enter your probability of success, number of trials, and target number of successes, and the calculator handles the rest.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"What Is the Binomial CDF?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The binomial cumulative distribution function (CDF) gives you the probability of getting ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"x or fewer successes","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" in a fixed number of independent trials, where each trial has the same probability of success.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Think of it this way: the binomial PDF tells you the probability of getting ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"exactly","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" 5 heads in 10 coin flips. The binomial CDF tells you the probability of getting ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"5 or fewer","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" heads in those same 10 flips. That cumulative view is often what you actually need when making decisions based on probability.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For a result to follow a binomial distribution, four conditions need to hold:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Fixed number of trials","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — You know in advance how many trials you'll run (n)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Two possible outcomes","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Each trial results in either a \"success\" or \"failure\"","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Constant probability","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — The probability of success (p) stays the same for every trial","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Independent trials","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — The outcome of one trial doesn't affect the others","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"These conditions show up more often than you might expect. Coin flips, pass/fail inspections, yes/no survey responses, free throw attempts, and defect rates in manufacturing all fit the binomial model.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Understanding the Binomial CDF Formula","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The binomial CDF is calculated by summing individual binomial probabilities from 0 up to x:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≤ x) = Σ from k=0 to x of [C(n, k) × p^k × (1 − p)^(n − k)]","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Where:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"n","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = total number of trials","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"x","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = number of successes you're evaluating","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"p","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = probability of success on a single trial","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"C(n, k)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = the binomial coefficient, or \"n choose k\" = n! / (k! × (n − k)!)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Each term in the sum, C(n, k) × p^k × (1 − p)^(n − k), gives the probability of getting exactly k successes. The CDF adds up all those individual probabilities from 0 through x.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Here's a quick worked example. Say you flip a fair coin 10 times and want to know the probability of getting 3 or fewer heads:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"n = 10, p = 0.5, x = 3","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"P(X ≤ 3) = 0.0010 + 0.0098 + 0.0439 + 0.1172","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≤ 3) = 0.1719","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"So there's about a 17.2% chance of getting 3 or fewer heads in 10 fair coin flips. This calculator does that summation instantly for any values you enter.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"How to Use This Calculator","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Enter the Probability of Success (p):","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Use the slider or type a value between 0 and 1. For a fair coin, this would be 0.5. For a process with a 90% pass rate, enter 0.9.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Enter the Number of Trials (n):","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Type the total number of independent trials. This might be the number of items inspected, questions on a test, or repetitions of an experiment.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Enter the Number of Successes (x):","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Type the specific number of successes you want to evaluate probabilities for.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Read Your Results:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The calculator displays five probabilities:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"number","start":1,"tag":"ol"},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X = x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Probability of ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"exactly","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" x successes","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X < x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Probability of ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"fewer than","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" x successes","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≤ x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Probability of x ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"or fewer","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" successes (this is the CDF)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X > x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Probability of ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"more than","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" x successes","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≥ x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" — Probability of x ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"or more","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" successes","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":5}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"All five values update automatically as you change your inputs.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Practical Examples","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Quality Control Inspection","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A manufacturer knows that 3% of items coming off a production line are defective. An inspector checks a batch of 50 items. What's the probability of finding 3 or fewer defective items?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"p = 0.03, n = 50, x = 3","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≤ 3) = 0.9372","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There's about a 93.7% chance the batch will have 3 or fewer defects. If the inspector finds 5 or more, that would be unusual enough to warrant investigating the production line.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Free Throw Shooting","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A basketball player has a 78% free throw percentage. In a game, she shoots 12 free throws. What's the probability she makes at least 10?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"p = 0.78, n = 12, x = 10","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≥ 10) = 0.5765","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"She has about a 57.7% chance of making 10 or more of her 12 free throws. Not a guarantee, but better than a coin flip.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Medical Clinical Trial","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A new treatment has a 60% response rate based on early data. In a trial of 20 patients, what's the probability that fewer than 10 respond?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"p = 0.60, n = 20, x = 10","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X < 10) = 0.1275","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There's roughly a 12.8% chance that fewer than 10 out of 20 patients will respond. If the trial shows fewer than 10 responders, it might raise questions about whether the real response rate is lower than expected.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Multiple Choice Guessing","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A student guesses randomly on a 15-question multiple choice quiz with 4 options per question. What's the probability of getting exactly 5 correct?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"p = 0.25, n = 15, x = 5","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X = 5) = 0.1651","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There's about a 16.5% chance of guessing exactly 5 right. And P(X ≥ 5) = 0.3135, meaning there's only a 31.4% chance of getting 5 or more correct by pure guessing.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Binomial CDF vs. PDF: What's the Difference?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This is one of the most common points of confusion in probability, so let's clear it up.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"type":"table","children":[{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":1,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Binomial PDF","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":1,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Binomial CDF","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":1,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1},{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"**What it calculates**","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Probability of *exactly* x successes","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Probability of *x or fewer* successes","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1},{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"**Formula notation**","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"P(X = x)","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"P(X ≤ x)","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1},{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"**Example**","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Probability of exactly 5 heads in 10 flips","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"Probability of 5 or fewer heads in 10 flips","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1},{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"**On a TI-84**","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"binompdf(n, p, x)","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"binomcdf(n, p, x)","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1},{"type":"tablerow","children":[{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"**When to use**","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"You need the probability of one specific outcome","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1},{"type":"tablecell","children":[{"type":"paragraph","children":[{"type":"text","text":"You need the probability of a range of outcomes","format":0,"style":"","detail":0,"mode":"normal","version":1}],"direction":"ltr","format":"","indent":0,"textFormat":0,"textStyle":"","version":1}],"direction":"ltr","format":"","indent":0,"version":1,"backgroundColor":null,"colSpan":1,"headerState":0,"rowSpan":1}],"direction":"ltr","format":"","indent":0,"version":1}],"direction":"ltr","format":"","indent":0,"version":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A helpful way to think about it: the PDF gives you the height of a single bar in the probability distribution, while the CDF gives you the total area of all bars from 0 up to x.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This calculator gives you both. P(X = x) is the PDF value, and P(X ≤ x) is the CDF value. The other outputs — P(X < x), P(X > x), and P(X ≥ x) — are derived from these two.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Here's how they relate:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X < x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = P(X ≤ x) − P(X = x)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X > x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = 1 − P(X ≤ x)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"P(X ≥ x)","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" = 1 − P(X < x) = 1 − P(X ≤ x) + P(X = x)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"When to Use Cumulative Binomial Probability","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Cumulative probabilities come up whenever your question involves \"at most,\" \"at least,\" \"fewer than,\" or \"more than\" — which is most real-world probability questions. Here are some common scenarios:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"\"At most\" questions:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" \"What's the probability of at most 3 defects?\" Use P(X ≤ 3).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"\"At least\" questions:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" \"What's the probability of at least 8 successes?\" Use P(X ≥ 8).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"\"Fewer than\" questions:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" \"What's the probability of fewer than 5 correct answers?\" Use P(X < 5).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"\"More than\" questions:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" \"What's the probability of more than 2 failures?\" Reframe in terms of successes and use the appropriate output.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Hypothesis testing:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Determining whether an observed result is statistically unusual given an expected probability.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":5},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Range probabilities:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" For P(a ≤ X ≤ b), calculate P(X ≤ b) − P(X ≤ a−1) by running the calculator twice.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":6}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"}],"direction":null,"format":"","indent":0,"type":"root","version":1}}},"id":"69ab0d8141075e0004c809d9"}]}]}],"$L35"]}],"$L36"]}]]}]}]}]
3c:I[51318,["9310","static/chunks/f8d55a4d-f82e876c81147bfd.js","6956","static/chunks/66a6e935-aee5d3cb85c323ac.js","7821","static/chunks/7821-fecf606f47f1f58d.js","8963","static/chunks/8963-69c253eec8d741c5.js","9858","static/chunks/9858-f1918ded0530ca69.js","609","static/chunks/609-95e41ae6a69b88b2.js","2456","static/chunks/2456-94cb89c116d69dd0.js","994","static/chunks/994-2c56ebb2d663b523.js","6472","static/chunks/6472-2ee935ddbae5821a.js","3349","static/chunks/app/(frontend)/calculators/%5Bcategory%5D/%5Bslug%5D/page-70bb6ad84377852d.js"],"Calculator"]
35:["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-8","children":[["$","h2",null,{"className":"typo-large mb-6 text-mist-950","children":"Frequently Asked Questions"}],["$","div",null,{"className":"space-y-6","children":[["$","div","0",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What does binomcdf stand for?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Binomcdf stands for \"binomial cumulative distribution function.\" It calculates the probability of getting x or fewer successes in n independent trials, each with the same probability of success p. If you've used a TI-84 calculator, the binomcdf function works the same way — this online calculator just makes it accessible from any device.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","1",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How do I calculate P(X >= x) using binomial CDF?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"You can get P(X ≥ x) from the CDF using the complement: P(X ≥ x) = 1 − P(X ≤ x − 1). But with this calculator, you don't need to do that math yourself. Just enter your values and read P(X ≥ x) directly from the results.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","2",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What's the difference between binomcdf and binompdf?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Binompdf calculates the probability of ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"exactly","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" x successes (a single point), while binomcdf calculates the probability of ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"x or fewer","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" successes (a cumulative total). For example, binompdf(10, 0.5, 3) gives the probability of exactly 3 heads in 10 coin flips, while binomcdf(10, 0.5, 3) gives the probability of 0, 1, 2, or 3 heads.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","3",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can I use this calculator for a range of values like P(3 <= X <= 7)?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Yes, but it takes two steps. Calculate P(X ≤ 7) and subtract P(X ≤ 2). The difference gives you P(3 ≤ X ≤ 7). Run the calculator once with x = 7 to get the upper bound, and once with x = 2 to get the value to subtract.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","4",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What if my probability of success is greater than 0.5?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The calculator works for any probability between 0 and 1. A high probability of success simply means successes are more likely than failures, which shifts the distribution to the right. The formulas and calculations work exactly the same way.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],"$L37","$L38","$L39","$L3a","$L3b"]}]]}]
36:["$","div",null,{"className":"lg:col-span-2","children":[["$","div",null,{"className":"sticky top-24 mb-6 max-h-[calc(100vh-6rem)] overflow-y-auto space-y-4","children":[["$","$L3c",null,{"className":"rounded-xl shadow-sm","id":"cmmesz1da000gl104298ix6l2"}],["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-4","children":["$","nav",null,{"aria-label":"Table of contents","children":[["$","h2",null,{"className":"text-lg font-display font-normal mb-4 text-mist-900","children":"Table of Contents"}],["$","ul",null,{"className":"space-y-2.5","children":[["$","li","what-is-the-binomial-cdf",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#what-is-the-binomial-cdf","children":"What Is the Binomial CDF?"}]}],["$","li","understanding-the-binomial-cdf-formula",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#understanding-the-binomial-cdf-formula","children":"Understanding the Binomial CDF Formula"}]}],["$","li","how-to-use-this-calculator",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#how-to-use-this-calculator","children":"How to Use This Calculator"}]}],["$","li","practical-examples",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#practical-examples","children":"Practical Examples"}]}],["$","li","quality-control-inspection",{"className":"ml-3","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#quality-control-inspection","children":"Quality Control Inspection"}]}],["$","li","free-throw-shooting",{"className":"ml-3","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#free-throw-shooting","children":"Free Throw Shooting"}]}],["$","li","medical-clinical-trial",{"className":"ml-3","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#medical-clinical-trial","children":"Medical Clinical Trial"}]}],["$","li","multiple-choice-guessing",{"className":"ml-3","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#multiple-choice-guessing","children":"Multiple Choice Guessing"}]}],["$","li","binomial-cdf-vs-pdf-whats-the-difference",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#binomial-cdf-vs-pdf-whats-the-difference","children":"Binomial CDF vs. PDF: What's the Difference?"}]}],["$","li","when-to-use-cumulative-binomial-probability",{"className":"","children":["$","$L6",null,{"className":"block py-1 text-mist-600 hover:text-mist-900 hover:underline transition-colors leading-relaxed text-sm no-underline","href":"#when-to-use-cumulative-binomial-probability","children":"When to Use Cumulative Binomial Probability"}]}]]}]]}]}]]}],["$","div",null,{"className":"space-y-6","children":false}]]}]
37:["$","div","5",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How many trials can this calculator handle?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The calculator handles a wide range of trial numbers. For very large values of n (say, over 1,000), the binomial distribution closely approximates a normal distribution, and you might find a normal approximation calculator useful as a cross-check. For most practical problems, this calculator will give you accurate results.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
38:["$","div","6",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Why are my results showing 0 or 1?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"If you see probabilities of 0 or very close to 0, it usually means the outcome you're asking about is extremely unlikely given your inputs. Similarly, a probability of 1 (or very close to it) means the outcome is nearly certain. Try adjusting your number of successes (x) closer to n × p (the expected value) to see more intermediate probabilities.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
39:["$","div","7",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"When should I use binomial CDF vs. the normal approximation?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For small to moderate sample sizes (n < 30 or so), use the exact binomial CDF — that's what this calculator provides. When n is large and p isn't too close to 0 or 1, the normal approximation with continuity correction becomes a reasonable shortcut. As a rule of thumb, the normal approximation works well when both n × p ≥ 5 and n × (1 − p) ≥ 5.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3a:["$","div","8",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What are common mistakes when using binomial CDF?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The most common mistakes are mixing up \"fewer than\" and \"at most\" (P(X < 5) vs. P(X ≤ 5) — they differ by exactly P(X = 5)), forgetting that trials must be independent, and using the binomial model when the probability changes between trials. Also watch for off-by-one errors: \"at least 5\" means P(X ≥ 5), not P(X > 5).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3b:["$","div","9",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Is this the same as the binomcdf function on a TI-84?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L32",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Yes, this calculator produces the same results as the TI-84's binomcdf function. On a TI-84, you'd access it through 2nd > VARS > binomcdf(n, p, x). This online version saves you from navigating calculator menus and gives you all five probability types at once, rather than just the cumulative P(X ≤ x).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]