A probability calculator for events, trials and odds
This free probability calculator answers the four kinds of chance question that come up most: the probability of a single event from favourable and total outcomes; how two events combine, whether independent or dependent, including conditional probability with Bayes’ theorem; the chance of k successes in repeated trials (the binomial distribution); and conversion between odds and probability in fractional, decimal and American formats.
Every answer shows the formula and the arithmetic. Fractions stay exact (½ × ⅙ is 1/12) and binomial tails use a log-space method that stays accurate for a million trials, where the usual formula overflows. Everything runs in your browser.
How to use the Probability Calculator
- Choose a tab: Single event, Two events, Repeated trials or Odds ↔ probability.
- Single event: enter the favourable and total outcomes, such as 4 aces out of 52 cards.
- Two events: type P(A) and P(B) (0.25, 25%, 1/4 or “1 in 4”) and say how they are related; dependent events also need P(A and B), P(B | A) or P(A | B).
- Repeated trials: enter n, p and k, then pick exactly, at most, at least, fewer, more or between.
- Odds: pick the format you have and type the value.
- Read the answer, the Venn diagram or chart and the numbered steps; “Copy result” copies the numbers.
Probability Calculator features
Exact fractions
Fractions and decimals are combined with exact rational arithmetic, so answers like 1/221 stay exact.
Dependent events and Bayes
Give P(A and B), P(B | A) or P(A | B); impossible combinations are caught with the valid range explained.
Accurate binomial tails
P(X = k), at most, at least and between are summed in log space, so n = 1,000,000 or a 10⁻⁵⁰ tail is still precise.
At least once and drop rates
Shows 1 − (1 − p)ⁿ, the tries needed for a 50, 90, 95 and 99% chance of at least one success, and the expected tries (1 ÷ p) — enter a drop rate as p.
Every odds format
Fractional, decimal, American and odds in favour, with the implied probability of each.
Probability formulas at a glance
| Question | Formula | Example |
|---|---|---|
| One event | P(A) = favourable ÷ total | An ace: 4/52 = 1/13 |
| Not A (complement) | P(not A) = 1 − P(A) | No six on one roll: 5/6 |
| A and B, independent | P(A) × P(B) | Heads and a six: ½ × ⅙ = 1/12 |
| A and B, dependent | P(A) × P(B | A) | Two aces in a row: 4/52 × 3/51 = 1/221 |
| A or B | P(A) + P(B) − P(A and B) | Heads or a six: ½ + ⅙ − 1/12 = 7/12 |
| A given B (Bayes) | P(B | A) × P(A) ÷ P(B) | Medical test, below |
| Exactly k in n trials | C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ | 5 heads in 10 flips: 252/1024 = 0.24609375 |
| At least once in n | 1 − (1 − p)ⁿ | A six in 4 rolls: 1 − (5/6)⁴ ≈ 51.77% |
Worked examples
Two aces in a row. Drawing without replacement makes the second draw depend on the first. P(first ace) = 4/52; once an ace is gone, P(second ace | first ace) = 3/51. Multiply: 4/52 × 3/51 = 12/2,652 = 1/221, about 0.45%. Choose “Dependent — I know P(B | A)” to reproduce it.
Bayes and a medical test. A condition affects 1% of people. A test detects it 90% of the time but also flags 5% of healthy people. A positive result has overall chance 0.9 × 0.01 + 0.05 × 0.99 = 5.85%, so P(condition | positive) = 0.009 ÷ 0.0585 ≈ 15.4%, far lower than most people guess.
Coin flips. Exactly 5 heads in 10 fair flips: 252 of 1,024 equally likely sequences, P = 0.24609375.
The “at least once” trap. A 1-in-6 chance tried 6 times is not certain: 1 − (5/6)⁶ ≈ 66.5%, and it passes 95% only after 17 rolls. The same maths gives a loot drop rate: a 1% drop needs 69 tries for even odds and 459 for 99%. Fixed pity systems differ; see the pity calculator.
Odds versus probability
Probability compares successes with all outcomes; odds compare successes with failures. A probability of 1/4 means odds of 1 : 3 in favour, or 3/1 against. Fractional 5/2 pays 5 for every 2 staked (implied 2 ÷ 7 ≈ 28.6%); decimal 3.50 includes the stake; American +250 is the profit on 100, −150 the stake to win 100.
Bookmakers’ prices include a margin, so implied probabilities sum to over 100%.
- Evens = 1/1 = decimal 2.00 = American +100 = 50%.
- To count the outcomes themselves — lottery tickets, poker hands — use the permutation and combination calculator; to simulate chance, try the dice roller or coin flip.
Probability Calculator FAQ
How do I use a probability calculator for two events?
Enter P(A) and P(B), then say whether the events are independent, mutually exclusive or dependent. For dependent events also give P(A and B) or a conditional probability. It returns P(A and B), P(A or B), exactly one, neither and both conditionals.
What is the difference between independent and mutually exclusive events?
Independent events don’t affect each other, so P(A and B) = P(A) × P(B). Mutually exclusive events cannot happen together, so P(A and B) = 0 — knowing one happened tells you the other didn’t.
How do I calculate the probability of at least one success?
Work out the chance of no successes and subtract it from 1: P(at least one) = 1 − (1 − p)ⁿ. For a 10% chance tried 5 times, that is 1 − 0.9⁵ ≈ 41%.
When should I use the binomial distribution?
When there is a fixed number of independent trials with the same chance of success and two outcomes — heads or tails, hit or miss. Cards drawn without replacement are only approximated.
How do I convert odds to a probability?
For fractional odds a/b against, P = b ÷ (a + b). For decimal odds, P = 1 ÷ odds. For American odds, P = 100 ÷ (A + 100) when positive and A ÷ (A + 100) when negative, using the number without its sign.
How do I work out a drop rate chance?
Use Repeated trials: n is your number of attempts, p the drop rate (1% or 1/100), k = 1 with “at least”. Tiles show the tries needed for 50, 90 and 99% confidence. Attempts are assumed independent; pity or rising rates need the pity calculator.
Why is an answer shown like 3.2 × 10⁻⁴⁵?
Very small probabilities are kept accurate with logarithms and shown in scientific notation.