OneTaskBy Tim King · Updated
Probability Calculator
Independent Events, Exactly One Success, Neither Occurring and Odds Over n Tries
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How the Probability Calculator works
This probability calculator combines two independent events. Enter each chance as a decimal, a percentage or a fraction, such as 0.25, 25% or 1/4, and see P(A and B), P(A or B), exactly one, neither and each complement, all as percentages with "1 in n" odds. Add a number of tries to get the chance A happens at least once.
Independent events do not affect each other: a coin flip has no memory of the dice roll before it. For those events, "and" means multiply and "or" means add, then remove the overlap so it is not counted twice. The "at least once" question catches people out most. A 1 in 6 chance does not become certain after 6 tries; it reaches only about 67%, and gamers, gamblers and testers all benefit from knowing that.
How the probability calculator works
The formula for both is P(A and B) = P(A) × P(B). For either, P(A or B) = P(A) + P(B) − P(A) × P(B). Neither is (1 − P(A)) × (1 − P(B)). The chance of A at least once in n tries is 1 − (1 − P(A))^n, and the tries for a 50% chance is the smallest n where that passes one half.
First example. Rolling a 6 (1/6) and flipping heads (50%) together is 1/6 × 1/2 = 1/12, about 8.33%.
Second example. At least one 6 in 10 rolls is 1 − (5/6)^10 = 83.85%, not 100%.
How many tries for a 50% chance at a 1% drop? 1 − 0.99^n passes 50% at n = 69.
Chance of at least one success in n independent tries
| Tries (n) | p = 1/6 (a die roll) | p = 1/10 | p = 1/100 |
|---|---|---|---|
| 1 | 16.67% | 10% | 1% |
| 2 | 30.56% | 19% | 1.99% |
| 3 | 42.13% | 27.1% | 2.97% |
| 4 | 51.77% | 34.39% | 3.94% |
| 5 | 59.81% | 40.95% | 4.901% |
| 6 | 66.51% | 46.86% | 5.852% |
| 7 | 72.09% | 52.17% | 6.793% |
| 8 | 76.74% | 56.95% | 7.726% |
| 9 | 80.62% | 61.26% | 8.648% |
| 10 | 83.85% | 65.13% | 9.562% |
| 12 | 88.78% | 71.76% | 11.36% |
| 15 | 93.51% | 79.41% | 13.99% |
| 20 | 97.39% | 87.84% | 18.21% |
| 25 | 98.95% | 92.82% | 22.22% |
| 30 | 99.58% | 95.76% | 26.03% |
| 40 | 99.93% | 98.52% | 33.1% |
| 50 | 99.99% | 99.48% | 39.5% |
| 69 | Over 99.99% | 99.93% | 50.02% |
| 70 | Over 99.99% | 99.94% | 50.52% |
| 100 | Over 99.99% | Over 99.99% | 63.4% |
| 200 | Over 99.99% | Over 99.99% | 86.6% |
| 300 | Over 99.99% | Over 99.99% | 95.1% |
| 459 | Over 99.99% | Over 99.99% | 99.01% |
| 500 | Over 99.99% | Over 99.99% | 99.34% |
Tips and pitfalls
- Only for independent events. Drawing cards without replacement changes the odds each time.
- No gambler's fallacy. Ten tails in a row does not make heads more likely next flip.
- Do not add for "at least once". Six tries at 1/6 is 66.5%, not 100%.
- Percent, decimal or fraction. 0.05, 5% and 1/20 are all read as the same chance.
FAQ
How do you calculate the probability of two events?
For independent events, multiply for "both" and use the addition rule for "either". With P(A) = 0.3 and P(B) = 0.5, both happen with probability 0.15 and at least one happens with 0.3 + 0.5 − 0.15 = 0.65. Enter both chances and the calculator shows every combination.
What is the probability of at least one success?
It is one minus the chance of failing every time: 1 − (1 − p)^n. For a 10% chance over 5 tries, 1 − 0.9^5 = 41%. The chance climbs quickly at first and then slowly, never quite reaching 100%. The table above shows how it grows for common odds.
How do you convert odds to a probability?
"1 in 8" is a probability of 1/8 = 12.5%. Betting odds of 3 to 1 against mean 1 win in 4 attempts, so 1/4 = 25%. Type the fraction straight into the calculator, and every result is shown both as a percentage and as "about 1 in n".
What is the difference between independent and mutually exclusive events?
Independent events can both happen without affecting each other, such as two coin flips. Mutually exclusive events cannot both happen, such as rolling a 2 and a 5 on one die. For mutually exclusive events, P(A or B) is simply P(A) + P(B), with no overlap to subtract.
Does a 1 in 100 chance happen in 100 tries?
Not guaranteed. The chance of at least one success in 100 tries is 1 − 0.99^100 ≈ 63.4%. You need 69 tries for a 50% chance, 299 for 95% and 459 for 99%. That is why rare game drops can take far longer than the stated odds suggest.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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