Probability Calculator
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Results
P(A and B) = 0.0833333, P(A or B) = 0.583333 for independent events with P(A) = 0.5 and P(B) = 0.166667
| Outcome | Probability | Percent |
|---|---|---|
| A and B | 0.0833333 | 8.33% |
| A only | 0.416667 | 41.67% |
| B only | 0.0833333 | 8.33% |
| Neither | 0.416667 | 41.67% |
These rules assume A and B are independent: one happening does not change the chance of the other. For dependent events, P(A and B) = P(A) × P(B | A).
How to use the probability calculator
- Choose a calculation: two events, repeated trials or a normal distribution.
- Enter probabilities as decimals (0.25), percentages (25%) or fractions (1/4). Fractions keep dice and card problems exact.
- For repeated trials, enter the chance per trial, the number of trials n and the number of successes k. A full table of the distribution appears below the results.
- For a normal distribution, enter the mean, standard deviation and one or both limits.
Probability rules and formulas
Independent events: P(A and B) = P(A) × P(B)
Addition rule: P(A or B) = P(A) + P(B) − P(A and B)
Exactly one: P(A xor B) = P(A) + P(B) − 2 P(A and B) · Neither: (1 − P(A)) × (1 − P(B))
Binomial: P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ · At least once: 1 − (1 − p)ⁿ
Normal: P(a < X < b) = Φ((b − μ) ÷ σ) − Φ((a − μ) ÷ σ)
A probability is a number from 0 (impossible) to 1 (certain). Two events are independent when one happening does not change the chance of the other, like separate coin flips. The binomial formula counts the C(n, k) ways to place k successes among n trials and multiplies by the chance of each arrangement. Φ is the standard normal cumulative distribution function. Cumulative binomial probabilities are computed with the regularized incomplete beta function, so they stay accurate even for millions of trials.
Worked examples
A coin and a die. Let A be "heads" (0.5) and B be "a six" (1/6). Then P(A and B) = 0.5 × 1/6 = 0.08333, P(A or B) = 0.5 + 1/6 − 1/12 = 0.5833, and the chance of neither is 0.5 × 5/6 = 0.4167.
The Chevalier de Méré's problem. In the 1650s this gambler's question helped prompt Pascal and Fermat's correspondence on probability. The chance of at least one six in four rolls of a die is 1 − (5/6)⁴ = 0.5177, a good bet; the chance of at least one double six in 24 rolls of two dice is 1 − (35/36)²⁴ = 0.4914, a bad one, even though both seem to follow the same pattern.
Exactly two sixes in ten rolls. C(10, 2) × (1/6)² × (5/6)⁸ = 45 × 0.02778 × 0.2326 = 0.2907. The chance of at least two is 0.5155.
A normal range. IQ scores are scaled to a mean of 100 and a standard deviation of 15. The share between 85 and 115 (z = −1 to 1) is 68.27%, and the share above 130 (z = 2) is 2.28%.
Chance of at least one success
Each cell is 1 − (1 − p)ⁿ, computed with this calculator's engine.
| Trials (n) | p = 1% | p = 5% | p = 1/6 (a die) | p = 50% (a coin) |
|---|---|---|---|---|
| 1 | 1% | 5% | 16.67% | 50% |
| 2 | 1.99% | 9.75% | 30.56% | 75% |
| 5 | 4.9% | 22.62% | 59.81% | 96.88% |
| 10 | 9.56% | 40.13% | 83.85% | 99.9% |
| 20 | 18.21% | 64.15% | 97.39% | > 99.9999% |
| 50 | 39.5% | 92.31% | 99.99% | > 99.9999% |
| 100 | 63.4% | 99.41% | > 99.9999% | > 99.9999% |
| Trials for a 50% chance | 69 | 14 | 4 | 1 |
| Trials for a 95% chance | 299 | 59 | 17 | 5 |
Even a 1% risk per attempt becomes likely with enough attempts: after 100 tries the chance it has happened at least once is 63.4%. This is why small per-use failure rates matter for things used many times.
Common mistakes
- Adding probabilities that overlap. P(A or B) is not P(A) + P(B) unless A and B cannot happen together. Heads or a six is 7/12, not 2/3.
- Confusing independent with mutually exclusive. Mutually exclusive events cannot both happen, so they are strongly dependent: if one happens, the other cannot.
- The gambler's fallacy. After five tails in a row, the next fair coin flip is still 50% heads. Independent trials have no memory.
- Multiplying dependent events. Drawing two aces without replacement is 4/52 × 3/51, not (4/52)², because the first draw changes the deck.
To count the outcomes behind a probability, such as lottery tickets or poker hands, use the permutation and combination calculator. For tail areas of the normal curve and percentiles, use the z-score calculator.
Frequently asked questions
How do you find the probability of A and B?
For independent events, multiply: P(A and B) = P(A) × P(B). The chance of flipping heads and rolling a six is 0.5 × 1/6 = 1/12, about 0.0833 or 8.33%.
How do you find the probability of A or B?
Add the two probabilities and subtract the overlap so it is not counted twice: P(A or B) = P(A) + P(B) − P(A and B). For heads or a six, 0.5 + 1/6 − 1/12 = 7/12, about 0.5833.
What is the chance of something happening at least once?
Work out the chance it never happens and subtract that from 1: P(at least once) = 1 − (1 − p)ⁿ. The chance of at least one six in four rolls of a die is 1 − (5/6)⁴ ≈ 0.5177, just better than even.
How do you find the probability of exactly k successes?
Use the binomial formula C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, which assumes independent trials with the same chance each time. The probability of exactly two sixes in ten rolls is C(10, 2) × (1/6)² × (5/6)⁸ ≈ 0.2907.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other, so P(A and B) = P(A) × P(B). Mutually exclusive events cannot happen together, so P(A and B) = 0 and P(A or B) = P(A) + P(B). Rolling a 1 and rolling a 6 on one die are mutually exclusive, not independent.