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Probability Calculator
Find the probability of A and B, A or B, not A and conditional probability for two events, or the chance that something happens at least once in a number of tries. Type probabilities as decimals, percents or fractions.
Calculate
Decimal (0.5), percent (50%) or fraction (1/2)
Decimal, percent or fraction
Probability that both happen
Decimal, percent or fraction
Binomial probability of exactly k successes
Enter your values to see the result.
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How to use the Probability Calculator
- Choose a mode: two events A and B, or at least once in n trials.
- Enter the probabilities as a decimal (0.25), a percent (25%) or a fraction (1/4).
- Pick how A and B relate: independent, mutually exclusive, or enter P(A and B) if you know it.
- Press Calculate probability to see every combined probability with the formula steps.
- Copy the results or download the table as CSV.
Probability formulas
| What you want | Formula | Example (P(A) = 0.5, P(B) = 0.2, independent) |
|---|---|---|
| Not A (complement) | P(A') = 1 − P(A) | 0.5 |
| A and B (independent) | P(A) × P(B) | 0.1 |
| A and B (dependent) | P(A) × P(B | A) | — |
| A or B | P(A) + P(B) − P(A and B) | 0.6 |
| Exactly one (xor) | P(A) + P(B) − 2·P(A and B) | 0.5 |
| Neither | 1 − P(A or B) | 0.4 |
| A given B | P(A and B) ÷ P(B) | 0.5 |
| At least once in n trials | 1 − (1 − p)n | — |
| Exactly k in n trials | C(n, k) · pk · (1 − p)n−k | — |
The overlap P(A and B) must be between max(0, P(A) + P(B) − 1) and min(P(A), P(B)); the calculator checks this when you enter it yourself.
Common probabilities for reference
| Event | Fraction | Decimal | Percent |
|---|---|---|---|
| Coin lands heads | 1/2 | 0.5 | 50% |
| Die shows a six | 1/6 | 0.1667 | 16.67% |
| Two dice total 7 | 6/36 = 1/6 | 0.1667 | 16.67% |
| Two dice both six | 1/36 | 0.0278 | 2.78% |
| Draw a heart from a 52-card deck | 13/52 = 1/4 | 0.25 | 25% |
| Draw an ace | 4/52 = 1/13 | 0.0769 | 7.69% |
| At least one six in 4 rolls | 1 − (5/6)4 | 0.5177 | 51.77% |
| At least one double six in 24 rolls of two dice | 1 − (35/36)24 | 0.4914 | 49.14% |
The last two rows are the classic problem of the Chevalier de Méré that led Pascal and Fermat to found probability theory: the first bet wins slightly more than half the time, the second slightly less.
Frequently asked questions
How do you calculate the probability of A and B?
For independent events, multiply: P(A and B) = P(A) × P(B). For example a coin landing heads (1/2) and a die showing six (1/6) gives 1/12, about 8.33%. For dependent events use P(A and B) = P(A) × P(B|A).
How do you calculate the probability of A or B?
Use the addition rule P(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap stops it being counted twice. For mutually exclusive events the overlap is 0, so the probabilities simply add.
What is conditional probability?
P(A|B) is the probability of A given that B has happened. It equals P(A and B) ÷ P(B). If A and B are independent, P(A|B) is the same as P(A).
How do I find the probability of something happening at least once?
Calculate the chance it never happens and subtract from 1: P(at least once) = 1 − (1 − p)^n. The chance of rolling at least one six in 4 rolls is 1 − (5/6)^4 ≈ 51.77%.
What is the difference between independent and mutually exclusive?
Independent events do not affect each other, so both can happen (P(A and B) = P(A) × P(B)). Mutually exclusive events cannot happen together (P(A and B) = 0). Two events with non-zero probabilities cannot be both.
Can a probability be greater than 1?
No. A probability is always between 0 (impossible) and 1 (certain), or 0% to 100%. If you type a whole number between 1 and 100, the calculator reads it as a percent.
How do I convert probability to odds?
The "1 in X" figure is 1 ÷ p. A probability of 0.04 is 1 in 25. Odds against are (1 − p) : p, so 0.04 is 24 to 1 against.
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