How to use the probability calculator
- Choose what you want to work out: two independent events, at least one success over several tries, or binomial probabilities for exactly, at most or at least k successes.
- Enter probabilities as decimals from 0 to 1. A 20% chance is 0.2; a fair coin landing heads is 0.5. If you type a percentage such as 50, the calculator tells you the decimal to use.
- For the other two modes, enter the number of tries and, for binomial, the number of successes you are asking about.
The probability rules
Two events are independent when one happening doesn't change the chance of the other, like two coin flips or two dice. p is the chance of success on one try, n the number of tries, k the number of successes, and C(n, k) is the number of ways to choose which k of the n tries succeed. The binomial mode assumes every try is independent with the same p. For "at most" and "at least", the calculator adds up the individual probabilities exactly, using the regularized incomplete beta function, so it stays accurate for up to a million tries.
Worked example
The calculator opens with P(A) = 0.5 and P(B) = 0.4, for example a coin landing heads and a 40% chance of rain:
- Both: 0.5 × 0.4 = 0.2, a 20% chance
- At least one: 0.5 + 0.4 − 0.2 = 0.7
- Exactly one: 0.5 + 0.4 − 2 × 0.2 = 0.5
- Neither: 0.5 × 0.6 = 0.3
The four outcomes (both, A only, B only, neither) are 0.2 + 0.3 + 0.2 + 0.3 = 1, as they must be.
With p = 0.2 and 10 tries, the chance of no successes is 0.8¹⁰ = 0.107374, so the chance of at least one is 1 − 0.107374 = 0.892626. The chance of exactly 3 successes is C(10, 3) × 0.2³ × 0.8⁷ = 120 × 0.008 × 0.2097152 = 0.201327.
Common results
| Question | Probability |
|---|---|
| Exactly 5 heads in 10 coin flips | 0.246094 |
| At least one six in 4 rolls of a die | 0.517747 |
| At least one double six in 24 rolls of two dice | 0.491404 |
| 60 or more heads in 100 flips | 0.028444 |
The two dice questions are a famous 17th-century puzzle. Gamblers expected them to have the same odds, but one is slightly better than even and the other slightly worse.
Mistakes to avoid
Adding probabilities for "or" works only when the events can't both happen. For a coin and the weather, 0.5 + 0.4 = 0.9 counts the "both" case twice, so subtract it once to get 0.7.
Multiplying for "and" works only for independent events. Drawing two aces from a deck without putting the first one back is not independent: the second draw has 3 aces left in 51 cards, not 4 in 52.
A 10% chance doesn't become certain after 10 tries. The chance of at least one success is 1 − 0.9¹⁰ = 65.1%, and even after 22 tries it is just over 90%.
Frequently asked questions
What does independent mean in probability?
Two events are independent if knowing that one happened tells you nothing about the other. Separate coin flips, dice rolls and spins of a wheel are independent. Drawing cards without replacement, or the weather on two days in a row, usually aren't.
How do I find the probability of at least one success?
Work out the chance of no successes at all, then subtract it from 1. With a 20% chance per try and 10 tries, no successes has a chance of 0.8¹⁰ = 0.107, so at least one success is 0.893, or about 89%.
What is the difference between "at most" and "at least"?
"At most 3" means 0, 1, 2 or 3 successes. "At least 3" means 3 or more. Both include exactly 3, which is why P(X ≤ 3) and P(X ≥ 3) add up to more than 1.
How is this related to combinations?
The binomial formula uses C(n, k), the number of combinations of k items from n. The permutation and combination calculator shows how it is calculated and handles very large values.
Related calculators
- Permutation and Combination Calculator — nPr and nCr, with or without repetition, as exact whole numbers even when they run to thousands of digits.
- Z-Score Calculator — The z-score and percentile of a value, the probability between two values, and the value at any percentile.
- Dice Roller — Roll any number of d4, d6, d8, d10, d12, d20, d100 or custom dice with a modifier, and see the odds of your total.
- Sample Size Calculator — How many survey responses you need for a given confidence level and margin of error, or the margin for your sample.
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