Probability Calculator

Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1 (or equivalently 0% to 100%). A probability of 0 means the event is impossible, a probability of 1 (100%) means it is certain, and a probability of 0.5 (50%) means the event is just as likely to happen as not. For a simple event, probability is calculated as the number of favorable outcomes divided by the total number of equally likely outcomes.

Independent events are events where the outcome of one does not affect the outcome of the other — for example, flipping a coin twice, since the first flip has no bearing on the second. Dependent events are events where one outcome changes the probability of the next — for example, drawing two cards from a deck without replacement, since removing the first card changes the odds for the second draw. This calculator's AND/OR mode assumes the two events are independent, which is the most common classroom and real-world scenario.

Combinations (nCr) count the number of ways to choose r items from a set of n items when order does not matter — for example, choosing 3 lottery numbers from 10 where {1,2,3} is the same as {3,2,1}. Permutations (nPr) count the number of ways to arrange r items from a set of n when order does matter — for example, awarding 1st, 2nd, and 3rd place medals, where the same 3 people in a different order counts as a different outcome. Because permutations count every ordering separately, nPr is always greater than or equal to nCr for the same n and r.

For two independent events, the probability of both A and B happening is P(A and B) = P(A) × P(B) — you multiply the individual probabilities. The probability of A or B happening (or both) is P(A or B) = P(A) + P(B) − P(A) × P(B), which uses the inclusion-exclusion principle to avoid double-counting the overlap where both events occur. For example, if P(A) = 0.5 and P(B) = 0.3, then P(A and B) = 0.15 and P(A or B) = 0.5 + 0.3 − 0.15 = 0.65.

For a fair six-sided die, each face has a 1-in-6 chance, so the probability of rolling any specific number is 1/6 ≈ 16.7%. For a standard 52-card deck, the probability of drawing a specific card (like the ace of spades) is 1/52 ≈ 1.9%, while the probability of drawing any ace is 4/52 ≈ 7.7% since there are 4 favorable outcomes out of 52 total outcomes. These are classic examples of the single-event formula: favorable outcomes divided by total outcomes.

The formula nPr = n! / (n−r)! starts from all n! ways to arrange every item, then divides out the (n−r)! arrangements of the items you did not select, leaving only the orderings of the r chosen items. The formula nCr = n! / (r! × (n−r)!) takes that same permutation count and further divides by r! to remove duplicate orderings of the r chosen items, since combinations do not care about order. For large n, computing full factorials can overflow, so this calculator multiplies and divides incrementally instead of computing n! directly.