Probability Calculator

Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). At its simplest it is the count of favourable outcomes divided by the total number of outcomes, which is why a coin flip gives P(heads) = 1/2 and a six-sided die gives P(rolling 6) = 1/6, or about 0.167. The same value can be written as a decimal (0.5), a fraction (½), a percentage (50%) or as odds. Converting between them is small arithmetic: odds against an event equal (1-p)/p, so a probability of 0.25 becomes odds of 3:1 against.

The tool covers simple probability, compound events, conditional probability, permutations and combinations, and it treats the two events you enter as independent. For two independent events, the probability of A and B both happening is P(A) × P(B). For mutually exclusive events, the probability of A or B is P(A) + P(B). Where the two can overlap, the shared ground has to be removed once, so P(A OR B) = P(A) + P(B) − P(A AND B). Forgetting that last subtraction is the most common way to end up with an inflated answer.

Multiplying works because each independent event leaves the others untouched. Two coin flips both landing heads is ½ × ½ = ¼, and three dice all showing 6 is 1/6 × 1/6 × 1/6 = 1/216, or 0.46%. The trap is the gambler's fallacy: previous results do not affect future independent events, so five heads in a row does nothing to the next flip, which is still 50%. Anyone betting against a run is relying on a memory the coin does not have.

Dependence changes the arithmetic to P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A has already happened. Drawing two aces from a deck without replacement is the standard illustration. The first draw is 4/52, the second is 3/51 because both the aces and the pack have shrunk, and the combined chance is 12/2652, roughly 0.45%. Put the first card back and the draws are independent again at 4/52 × 4/52, or 0.59%. Conditional probability of this kind sits behind medical testing, weather forecasting and machine learning.

Counting selections is the other half of the subject. Combinations (nCr) count selections where order does not matter, such as lottery draws, while permutations (nPr) count arrangements where order matters. Both come from the factorial formula, and results are displayed as whole numbers, in full up to 15 digits. The birthday problem shows how badly intuition copes once pairs are involved. In a room of 23 people the chance that at least two share a birthday is 50.7%, rising to 97% with 50 people and 99.9% with 70, because you are not weighing 23 against 365 but counting the 253 pairs those people make.

A UK school class of 30 therefore carries a 70.6% chance of a shared birthday, though the result does not transfer to one specific date such as your own, where the odds are much longer. Cryptographers use the same maths in the birthday attack on hash functions, where a collision in an n-bit hash needs only about √(2^n) attempts.

UK gambling odds are worth seeing written down. The Lotto jackpot (6 numbers from 59) is 1 in 45,057,474, although any prize at all is 1 in 9.3. EuroMillions (5 from 50 plus 2 from 12) puts the jackpot at 1 in 139,838,160, and the Set for Life top prize at 1 in 15,339,390. Premium Bonds pay £1M at 1 in 60 billion per bond per month. Casino games are steadier but still tilted: European roulette with a single zero has a house edge of 2.7%, the American double-zero wheel 5.26%, blackjack played to basic strategy 0.5-1%, and bookmakers build in a margin of roughly 5-10%. Over the long run the house wins, because mathematical expectation is negative for the player on all of them.