Probability Calculator — Binomial, Normal, Poisson

Compute probabilities for binomial, normal, Poisson, and other distributions.

About this calculator

A probability calculator handles common probability problems: permutations and combinations (counting arrangements), conditional probability, Bayes' theorem, and discrete probability distributions (binomial, Poisson). Essential for combinatorics homework, gambling odds, quality control, and any kind of risk analysis.

The fundamental rule: P(A and B) = P(A) × P(B | A). When events are independent, P(B | A) = P(B). When dependent, conditional probability changes the second factor. Confusing the two is the #1 source of probability errors.

Formula

_nP_k = \frac{n!}{(n-k)!} \;;\; _nC_k = \frac{n!}{k!(n-k)!}

  • nPk — Permutations: arrangements where order matters
  • nCk — Combinations: selections where order doesn't matter
  • Note — nPk = k! × nCk

Worked example

Inputs: How many 5-card poker hands are possible from a 52-card deck?

  1. Order doesn't matter → use combinations
  2. 52C5 = 52! / (5! × 47!)
  3. = (52 × 51 × 50 × 49 × 48) / 120
  4. = 2,598,960 distinct hands

Frequently asked questions

What is the gambler's fallacy?

The mistaken belief that past random events affect future ones. After 5 consecutive reds on a roulette wheel, the next spin still has the same 18/37 probability of red — the wheel has no memory. Casinos profit from this fallacy when bettors increase wagers after losses, expecting a "due" win. Independent events stay independent.

How does Bayes' theorem work?

Bayes' theorem updates the probability of a hypothesis given new evidence: P(H|E) = P(E|H) × P(H) / P(E). Classic example: a disease has prevalence 1% (P(H)). A test is 95% accurate (P(E|H) = 0.95). If positive, the probability you actually have the disease is only ~16% — counter-intuitive because the false-positive rate (5%) is multiplied by the much larger healthy population. Bayes is the foundation of medical screening, spam filtering, and ML inference.

Related calculators

Descriptive Stats Hypothesis Testing Distributions

More calculators

EMI Calculator Mortgage Calculator SIP Calculator Compound Interest GST / VAT Calculator Salary Calculator Unit Converter Currency Converter Scientific Calculator Concrete Calculator Paint Calculator Tile Calculator Calorie / TDEE Body Fat Macro Calculator O-Ring Sizes Bolt & Nut Sizes Pipe Sizes Wire Gauge AWG Needle Gauge Chart IV Cannula Sizes Suture Sizes Catheter French Sizes Drug Dosage See all 90+ calculators ›