Probability Calculator

Calculate probability for basic events, combinations, permutations, dice, cards and coins

Basic Probability

Example: rolling a number ā‰¤ 3 on a 10-sided die ā†’ 3 favourable, 10 total

// Formula Reference

Basic probabilityP = favourable / total
Complement ruleP(not A) = 1 āˆ’ P(A)
Combinations C(n,r)n! / (r! Ɨ (nāˆ’r)!)
Permutations P(n,r)n! / (nāˆ’r)!
Binomial (coin)C(n,k) Ɨ p^k Ɨ (1āˆ’p)^(nāˆ’k)
Odds for : againstP : (1āˆ’P)

Understanding Probability

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It can also be expressed as a percentage (0ā€“100%) or as odds (e.g. 3:1 against).

Key Formulas

Combinations vs Permutations

Combinations count selections where order doesn't matter. Choosing 3 people from 10 for a committee ā€” {A,B,C} is the same as {C,A,B}. Permutations count arrangements where order matters. Choosing a president, vice-president and secretary from 10 people ā€” {A,B,C} and {C,A,B} are different outcomes.

Note: This calculator assumes fair, unbiased events. Card probabilities assume a standard 52-card deck with no jokers, drawing without replacement.
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Probability Explained
What probability means, how to calculate it, and why the Gambler's Fallacy catches everyone out.
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// 0 to 1 Scale

Probability is always between 0 (impossible) and 1 (certain). 0.5 = 50% chance. Anything above 1 or below 0 indicates an error.

// Complement Rule

It's often easier to calculate P(not happening) and subtract from 1. P(at least one 6 in 3 rolls) = 1 āˆ’ P(no 6 in 3 rolls).

// Independent Events

For independent events, multiply probabilities. P(two heads) = P(head) Ɨ P(head) = 0.5 Ɨ 0.5 = 0.25.

// Gamblers Fallacy

Past outcomes don't affect future independent events. After 10 heads in a row, the next flip is still 50/50.