Probability calculator
Choose the kind of question, enter probabilities as decimals, percentages or fractions, and read the result as a decimal and a percentage. Events A and B are treated as independent.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Pick the question from the drop-down.
- Enter the probability or probabilities. Fractions such as 3/10 and percentages such as 30% are accepted.
- For the trial-based questions, add the number of trials n and, for binomial, the number of successes k.
How it works
All values are probabilities between 0 and 1, shown with up to 10 significant digits. Single events report the complement and odds. A and B and A or B use the independent-event formulas. At least one uses 1 − (1 − p)^n. The binomial rows sum the exact probability mass from the relevant end, so tiny tail probabilities do not lose precision to subtraction.
Worked example
Ten fair coin flips (n = 10, p = 0.5, k = 3):
- P(X = 3) = 120 / 1024 = 0.1171875
- P(X ≤ 3) = (1 + 10 + 45 + 120) / 1024 = 0.171875
- P(X ≥ 3) = 1 − 56/1024 = 0.9453125
- Expected heads: 5.
Tips
- Trials must be independent with the same p each time. Drawing cards without replacement is not binomial.
- “At most k” is P(X ≤ k). “More than k” is P(X > k).
- The binomial questions accept n up to 1,000.
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). Two fair coins both landing heads is 0.5 × 0.5 = 0.25. This tool assumes independence; if one event changes the other's chances you need conditional probability instead.
How do you find the probability of A or B?
Add and subtract the overlap, which is P(A) + P(B) − P(A and B). For independent events that is P(A) + P(B) − P(A) × P(B). With 0.2 and 0.3 it is 0.2 + 0.3 − 0.06 = 0.44.
How do you calculate "at least one" in n trials?
Take the complement. The chance of no success in n independent trials is (1 − p)^n, so at least one is 1 − (1 − p)^n. Rolling a six at least once in 4 rolls is 1 − (5/6)^4 = 0.5177, about 51.8%.
What is the binomial probability?
For n independent trials with success chance p each, P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Exactly 3 heads in 10 fair coin flips is 120 × 0.5^10 = 0.1171875. The cumulative P(X ≤ k) adds this up from 0 to k.
What do odds mean here?
Odds in favor compare the chance of the event with the chance of it not happening, p : (1 − p). A probability of 0.3 gives odds of 0.4286 : 1, or about 3 : 7.