Probability Calculator

Probability is favorable over total; odds are favorable against unfavorable. Drawing a heart from a deck: probability 13/52 = 25%, but odds 13:39, i.e. 1:3 for. The two get conflated constantly. '3 to 1 odds' is a 25% probability, not 33%. Converting: odds of a:b means probability a/(a+b). Gambling and horse racing speak odds; science and statistics speak probability; this calculator shows both so the translation is visible.

Multiply for 'and' (both events happen): P(A and B) = P(A)×P(B), for independent events. Add for 'or' (at least one happens), then subtract the overlap so it isn't double-counted: P(A or B) = P(A) + P(B) − P(A)×P(B). Forgetting the subtraction is the classic error, and it shows itself when probabilities sum past 100%. Two 70% events don't have a 140% chance of at least one occurring; they have 91%.

Independent means one event's outcome doesn't change the other's probability. Separate coin flips, different dice. The formulas here assume it, and they break for dependent events: drawing two aces without replacement isn't (4/52)², because the first draw changes the deck. It's (4/52)×(3/51). The tell for dependence is 'without replacement' or any shared pool. Dependent problems need conditional probability, which means tracking how the situation changes step by step.

One question: does order matter? Choosing 3 pizza toppings from 10, order irrelevant, combinations: 10C3 = 120. Awarding gold, silver, bronze among 10 runners. Order is the whole point, permutations: 10P3 = 720. The relationship is exact: nPr = nCr × r!, because each unordered group of r can be arranged r! ways. Passwords and PINs are permutations (with repetition, actually, nʳ); lottery tickets and committees are combinations.