Work out the probability of a single event or two events together — both, at least one, or neither — as a fraction, a decimal and a percentage. Runs entirely in your browser — your numbers never leave this device.
| Sum | Ways (out of 36) | Probability |
|---|---|---|
| 2 | 1 | 1/36 ≈ 2.78% |
| 3 | 2 | 2/36 ≈ 5.56% |
| 4 | 3 | 3/36 ≈ 8.33% |
| 5 | 4 | 4/36 ≈ 11.11% |
| 6 | 5 | 5/36 ≈ 13.89% |
| 7 | 6 | 6/36 ≈ 16.67% |
| 8 | 5 | 5/36 ≈ 13.89% |
| 9 | 4 | 4/36 ≈ 11.11% |
| 10 | 3 | 3/36 ≈ 8.33% |
| 11 | 2 | 2/36 ≈ 5.56% |
| 12 | 1 | 1/36 ≈ 2.78% |
Every probability on this page is a comparison: the number of outcomes you want divided by the number of outcomes there could be, assuming each outcome is equally likely. Drawing a heart from a full deck is 13 favourable cards out of 52 possible cards, 13/52, which simplifies to 1/4 and is reported as a fraction, a decimal (0.25) and a percentage (25%) together, because different courses and industries expect different forms. A probability of 0 means the event cannot happen, 1 means it is certain, and everything real sits in between.
Two events need one question answered first: are they independent, or mutually exclusive? Independent events do not affect each other — two coin flips, or a die rolled twice — so the probability of both is the product of the separate probabilities, the probability of neither is the product of the two failure probabilities, and 'at least one' is 1 minus that. Mutually exclusive events cannot both happen at all — a card cannot be both a heart and a spade — so the probability of either is simply the sum. Mixing the two rules up is the most common probability error, which is why the calculator asks which situation you have rather than guessing. Events that change each other's odds, like drawing a second heart after the first was kept out of the deck — (13/52) × (12/51) = 1/17 ≈ 5.88% — are dependent, and the honest route there is to enter the adjusted second probability yourself.
Dice show why counting outcomes beats intuition. Two dice have 36 equally likely ordered outcomes, but the eleven possible sums are not equally likely at all: there is only one way to make 2 (1+1) and six ways to make 7. The full distribution is in the table below — 7 is the most likely sum at 6/36 ≈ 16.67%, and the probabilities fall away symmetrically on either side. Any dice question can be answered by counting the rows of that table that match it.
Probability compares favourable outcomes with all outcomes; odds compare favourable outcomes with unfavourable ones. Drawing a heart has probability 13/52 = 1/4, but odds of 13 to 39 — which simplify to 1 to 3 — because there are 39 cards that are not hearts. Bookmakers quote odds; school mathematics usually asks for probability. This calculator works in probability.
No. Favourable outcomes are a part of total outcomes, so the fraction can never exceed 1 (every outcome is favourable — the event is certain) and can never fall below 0 (no outcome is favourable — the event is impossible). A calculation that produces 1.3 or −0.2 has gone wrong somewhere, usually by adding probabilities for events that are not mutually exclusive.
Independent means one event happening does not change the other's chances — two separate coin flips. Mutually exclusive means the events cannot both happen at all — a single card draw being a heart or a spade. The formulas are opposites in spirit: both-happening is a product for independent events (and zero for mutually exclusive ones), while either-happening is a sum only for mutually exclusive events.
Go through the back door: work out the probability that neither happens and subtract from 1. For at least one head in two coin flips, P(neither) = 1/2 × 1/2 = 1/4, so P(at least one head) = 1 − 1/4 = 3/4 = 75%. This complement trick is far easier than adding up all the ways 'at least one' can occur, and it works for any number of independent events.
Because the sums are built from pairs, and pairs are the equally likely outcomes — there are 36 of them. A sum of 7 can be built six different ways, but a sum of 2 or 12 only one way each, so 7 is six times as likely as 2: 16.67% against 2.78%. The same counting principle explains most 'surprising' results with cards, coins and lottery draws.
No. It runs 100% in your browser — the counts and probabilities you enter are processed on your own device, nothing is uploaded or saved, and there is no record of your calculation after you close the page.
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