Quick answer: If P(A) = 0.5 and P(B) = 0.4 and the events are independent, P(A and B) = 0.5 × 0.4 = 0.2 and P(A or B) = 0.5 + 0.4 − 0.2 = 0.7.
How it's calculated
For independent events (one does not change the chance of the other), the multiplication rule gives P(A and B) = P(A) × P(B). The addition rule gives P(A or B) = P(A) + P(B) − P(A and B); the overlap is subtracted so it is not counted twice.
The complement of an event is everything else: P(not A) = 1 − P(A). "Neither" is (1 − P(A))(1 − P(B)), and the chance of A at least once in n tries is 1 − (1 − P(A))ⁿ.
These rules assume independence. Drawing cards without putting them back, for example, makes events dependent and needs conditional probability. For counting outcomes, see the permutation and combination calculator.
Worked examples
- P(A) = 0.5, P(B) = 0.4: and 0.2, or 0.7, neither 0.3
- Two dice, both sixes: 1/6 × 1/6 = 0.027778
- P(A) = 0.5 at least once in 3 tries: 1 − 0.5³ = 0.875
Frequently asked questions
What does independent mean?
Two events are independent when knowing one happened does not change the probability of the other, like two separate coin tosses.
What if the events are mutually exclusive?
Mutually exclusive events cannot both happen, so P(A and B) = 0 and P(A or B) = P(A) + P(B). Such events are not independent (unless one has probability 0), so do not use this calculator's "and" result for them.
Can I enter percentages or fractions?
Yes. 0.4, 40% and 2/5 all mean the same probability.
What's the chance of at least one six in four rolls?
A little over half, about 51.8%. The chance of no six in four rolls is (5/6)⁴, around 48.2%, and you take that away from 100%.
After five tails in a row, is heads more likely?
No. A fair coin doesn't remember, so the next toss is still 50/50. Believing a result is "due" is known as the gambler's fallacy.
How do I turn odds into a probability?
Odds of 3 to 1 against mean one win for every three losses, so the chance is 1 in 4, or 25%.