Probability Calculator
Find P(A and B), P(A or B), P(not A) and conditional probabilities, the chance of at least one success in n tries, and binomial P(X = k), P(X ≤ k) and P(X ≥ k).
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Probability Calculator?
This probability calculator handles the three questions that come up most. Two events: give P(A) and P(B), say whether they are independent, mutually exclusive or linked by a known joint or conditional probability, and it returns P(A and B), P(A or B), P(not A), P(A given B), P(B given A), exactly one and neither, with a Venn diagram. Repeated tries: the chance of at least one success in n attempts, and how many attempts you need for a 50% or 95% chance. Binomial: the probability of exactly k, at most k, fewer than k, at least k or more than k successes in n trials.
Probabilities can be typed as decimals (0.25), percentages (25%) or fractions (1/4), and impossible combinations are caught with a plain explanation, such as a joint probability larger than either event on its own.
- Two-event rules for independent, mutually exclusive, joint and conditional inputs
- Venn diagram of the four regions
- At least one success in n tries, with tries needed for 50% and 95%
- Binomial exactly, at most, less than, at least and more than
- Distribution chart with the included outcomes highlighted
- Accepts decimals, percentages and fractions
How to use the Probability Calculator
- Choose Two events, At least one in n tries, or Binomial.
- For two events, enter P(A) and P(B) and pick how they are related. If you know P(A and B) or P(B given A), choose that option and enter it.
- For repeated tries, enter the chance of success on one try and the number of tries.
- For binomial, enter the number of trials n, the success probability p and the number of successes k, then read the exactly, at most and at least results.
- Check the steps and the diagram or chart, then copy the result or share the link.
Worked examples
A coin and a die
Input
P(A) = 0.5 (heads), P(B) = 1/6 (a six), independent
Output
P(A and B) = 0.083333, P(A or B) = 0.583333, P(neither) = 0.416667
Independent events multiply: 1/2 x 1/6 = 1/12. The union adds them and removes the overlap once: 1/2 + 1/6 - 1/12 = 7/12.
Dependent events from a conditional
Input
P(A) = 0.3, P(B) = 0.4, P(B | A) = 0.5
Output
P(A and B) = 0.15, P(A | B) = 0.375, P(A or B) = 0.55
Knowing B raises the chance of A from 0.3 to 0.375, so the events are not independent.
De Méré's bet
Input
p = 1/6, n = 4 tries
Output
At least one six: 51.7747%; 4 tries for a 50% chance, 17 for 95%
The complement, no sixes, is (5/6)⁴ = 48.23%.
Three sixes in ten rolls
Input
Binomial n = 10, p = 1/6, k = 3
Output
P(X = 3) = 0.155045, P(X ≤ 3) = 0.930272, P(X ≥ 3) = 0.224773
The expected number of sixes is 10/6 = 1.67, so three or more happens a bit more than one time in five.
Background and insight
The rules are short. The complement rule says P(not A) = 1 - P(A). The addition rule says P(A or B) = P(A) + P(B) - P(A and B), where the subtraction stops the overlap being counted twice; for mutually exclusive events the overlap is zero. The multiplication rule says P(A and B) = P(A) x P(B | A), which reduces to P(A) x P(B) when A and B are independent. Conditional probability is P(A | B) = P(A and B) / P(B). The calculator checks the Fréchet bounds, max(0, P(A) + P(B) - 1) ≤ P(A and B) ≤ min(P(A), P(B)), so inputs that cannot exist are rejected.
"At least one" is easiest through its complement. If each try succeeds with probability p independently, the chance of no successes in n tries is (1 - p)^n, so P(at least one) = 1 - (1 - p)^n. This is the calculation behind the 17th-century gambling puzzle posed by the Chevalier de Méré: at least one six in four rolls of a die has probability 1 - (5/6)⁴ = 51.77%, just better than even, while at least one double six in 24 rolls of two dice is only 49.14%. The tool uses log1p and expm1 so tiny probabilities over many trials keep their precision.
The binomial distribution counts successes in a fixed number n of independent trials with the same probability p: P(X = k) = C(n, k) p^k (1 - p)^(n - k). Its mean is np and its standard deviation √(np(1 - p)). Cumulative probabilities add up the individual terms, and the chart highlights the ones included.
Common use cases
Statistics courses
Check answers to addition-rule, conditional-probability and binomial exercises and see the working.
Games and gambling
Work out the odds of rolling, drawing or hitting at least once over several turns.
Reliability and risk
Estimate the chance that at least one of n independent components fails, or that none do.
Quality control
Find the probability of k or more defective items in a sample when the defect rate is known.
Tips and limitations
- Independent and mutually exclusive are different things. Mutually exclusive events with non-zero probabilities are always dependent, because one happening rules the other out.
- Adding probabilities only works for mutually exclusive events. Otherwise subtract the overlap.
- The repeated-tries and binomial modes assume every trial is independent with the same p. Drawing cards without replacement breaks that assumption.
- P(A | B) and P(B | A) are usually different. Confusing them is the classic error in reading medical test results.
- For "at least k" in a binomial question, use P(X ≥ k), which includes k itself; "more than k" excludes it.
Frequently asked questions
How do you calculate the probability of A and B?
Multiply P(A) by P(B given A). If the events are independent this is simply P(A) x P(B): the chance of heads and a six is 0.5 x 1/6 = 1/12.
How do you calculate the probability of A or B?
Add the two probabilities and subtract the overlap: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events the overlap is zero, so you just add.
How do you find the probability of at least one success?
Subtract the probability of no successes from 1. With independent tries of probability p, that is 1 - (1 - p)^n. A 1% chance per try gives a 63.4% chance of at least one success in 100 tries.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other, so P(A and B) = P(A) x P(B). Mutually exclusive events cannot happen together, so P(A and B) = 0. If both have non-zero probability they cannot be both independent and mutually exclusive.
What is a binomial probability?
It is the chance of exactly k successes in n independent trials that each succeed with probability p: C(n, k) p^k (1 - p)^(n - k). Five heads in ten fair coin tosses has probability 252/1024 = 0.2461.
What is conditional probability?
The probability of A given that B has happened, written P(A | B) and equal to P(A and B) / P(B). It narrows the sample space to the cases where B is true.
Is there an in-depth guide to the Probability Calculator?
Yes. “Probability Rules, Permutations and Combinations” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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