Probability Calculator

Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). This calculator handles simple probability, compound events (AND/OR), conditional probability and permutations and combinations.

For independent events, the probability of A AND B is P(A) × P(B). For mutually exclusive events, the probability of A OR B is P(A) + P(B). For non-mutually exclusive events, P(A OR B) = P(A) + P(B) − P(A AND B). The calculator selects the correct formula based on your inputs.

Combinations (nCr) count selections where order does not matter, such as lottery draws. Permutations (nPr) count arrangements where order matters. The calculator computes both using the factorial formula and displays results as a whole number and as a probability.

Basic probability formula. Probability = (favourable outcomes) ÷ (total outcomes). Coin flip: P(heads) = 1/2. Six-sided die: P(rolling 6) = 1/6 ≈ 0.167. Lottery: 6 numbers from 59 = 1 in 45,057,474 (UK Lotto jackpot odds). Probability ranges 0 (impossible) to 1 (certain). Often expressed as decimal (0.5), fraction (½), percentage (50%), or odds (1:1). Convert: probability p → odds against = (1-p)/p. P=0.25 → odds 3:1 against.

Independent events — multiply. Two independent events both occurring: multiply probabilities. P(A and B) = P(A) × P(B). Two coin flips both heads: ½ × ½ = ¼. Three dice all showing 6: 1/6 × 1/6 × 1/6 = 1/216 (0.46%). Hot streak fallacy: previous results don't affect future independent events. Just because heads came up 5 times doesn't make tails more likely next flip — still 50%.

Dependent events and conditional probability. P(A and B) = P(A) × P(B|A) where P(B|A) is probability of B given A occurred. Drawing 2 aces from deck without replacement: P(1st ace) = 4/52; P(2nd ace given 1st was ace) = 3/51. Total: 4/52 × 3/51 = 12/2652 ≈ 0.45%. Cards with replacement: independent: 4/52 × 4/52 = 0.59%. Used in medical testing, weather forecasting, machine learning.

Birthday paradox — the surprising result. In room of 23 people, probability that 2+ share birthday = 50.7%. With 50 people: 97%. With 70: 99.9%. Mathematical reason: not 23/365 but pairs — 23 people = 253 pairs. UK school class of 30: 70.6% chance of shared birthday. Doesn't apply to a SPECIFIC date (e.g. your birthday) — that's much lower. Used in cryptography (birthday attack on hash functions): only need √(2^n) attempts to find collision in n-bit hash.

UK lottery and gambling odds. Lotto (6/59): jackpot 1 in 45,057,474; any prize 1 in 9.3. EuroMillions (5/50 + 2/12): jackpot 1 in 139,838,160. Set for Life: 1 in 15,339,390 top prize. Premium Bonds: 1 in 60 billion per bond per month for £1M. Casino roulette (single zero, European): house edge 2.7%; double zero (American) 5.26%. Blackjack basic strategy: house edge 0.5-1%. Sports betting: bookmaker margin ~5-10%. Long-term: house always wins. Mathematical expectation negative for player on all casino games.