Probability is a share of the possibilities
A probability answers one question: out of all the equally likely things that could happen, what share are the ones you want? It is written as a decimal between 0 and 1, as a percentage between 0% and 100%, or described with odds — three spellings of the same idea, all provided by this calculator. Five modes cover the questions people actually ask: a single event, both of two events, either of two events, at least once across repeated trials, and converting odds into a probability. Each field label states which mode uses it, so you only need the fields for the mode you picked.
The five formulas
The AND rule multiplies because both hurdles must be cleared in sequence: half the time A happens, and half of those times B happens too, giving a quarter. The OR rule adds but subtracts the overlap, because the cases where both happen were counted in both probabilities. The at least once rule works backwards through the complement: it is easier to calculate the chance of failing every trial and subtract from 1. The odds conversion treats odds a : b as a parts in favour out of a + b equal parts in total.
Independence is the hidden assumption
The AND, OR and at-least-once formulas assume the events are independent — the outcome of one does not change the chances of another. Two separate coin flips are independent; drawing two cards from one deck without putting the first back is not, because the deck has changed. Real life sits between: weather on consecutive days, for example, is correlated, so multiplying daily rain chances understates the chance of a wet week. Use these modes for coins, dice, draws with replacement, and roughly independent events, and treat dependent situations with care.
Why at least once surprises people
An event with a 50% chance per trial does not reach certainty in two trials — it reaches 75%, then 87.5%, then 93.75%, halving the remaining gap each time. With the defaults here (50% over 5 trials) the chance of at least one success is 96.875%. The same logic explains why a 1% daily risk becomes serious over a year: 1 − 0.99365 is about 97.4%. Small repeated chances accumulate much faster than intuition suggests, in both directions — lotteries and hazards alike.
Worked example
Rolling a fair die, the probability of a six is 1 ÷ 6 = about 16.67%. Flipping two coins and getting both heads is 50% × 50% = 25% in AND mode.
How to use this calculator
- Choose the mode that matches your question — each field label says which mode uses it.
- For a single event, enter favourable and total outcomes. For combined events, enter the percentages for A and B.
- For at least once, enter the single-trial percentage as Event A and the number of trials.
- For odds mode, enter the odds for and against as a : b.
- Read the probability as a percentage, a decimal and an approximate 1-in-N figure.
Odds and probability are not the same words
Bookmakers and newspapers say odds where mathematicians say probability, and the numbers differ. Odds of 3 : 7 against-style in favour form (3 for, 7 against) mean 3 favourable parts out of 10 in total — a 30% probability, not 3/7 = 42.9%. The odds mode on this page does that conversion properly. The single-event mode also runs it in reverse, showing the odds in favour that correspond to your favourable and total counts, simplified like a ratio. If you are comparing odds from different sources, convert them all to percentages first with this tool; percentages can be compared at a glance, odds cannot.