Many probability questions combine two events: rain on Saturday or Sunday, a part that is both scratched and dented, a customer who buys coffee given that they bought a pastry. This calculator applies the three core rules (complement, addition and multiplication) to two events, works out conditional probabilities in both directions, handles independent, mutually exclusive and overlapping events, and draws a Venn diagram with the probability of every region.
How to use the probability calculator
- Choose whether to enter decimals (0.3) or percentages (30).
- Enter P(A) and P(B).
- Tell the calculator how the events relate: independent, mutually exclusive, a known P(A and B), or a known P(A | B).
- Optionally enter a number of repetitions to see the chance that A happens at least once, every time or never.
- Read P(A or B) and P(A and B) on the tape, along with complements, “exactly one”, “neither” and both conditional probabilities.
Probability rules
From these, P(neither) = 1 − P(A or B) and P(exactly one) = P(A or B) − P(A and B).
Worked examples
Independent events. The forecast gives a 30% chance of rain on Saturday and 40% on Sunday, and you treat the days as independent.
- Both days: 0.3 × 0.4 = 0.12
- At least one rainy day: 0.3 + 0.4 − 0.12 = 0.58
- A dry weekend: 1 − 0.58 = 0.42
- Rain on exactly one day: 0.58 − 0.12 = 0.46
Overlapping, dependent events. At a café, 30% of customers buy coffee (A), 40% buy a pastry (B) and 15% buy both. Then P(A or B) = 0.3 + 0.4 − 0.15 = 0.55, and P(pastry | coffee) = 0.15 ÷ 0.3 = 0.5. Since 0.15 differs from 0.3 × 0.4 = 0.12, the purchases are not independent: coffee buyers are more likely than average to add a pastry.
Mutually exclusive events. Drawing one card, P(heart) = 0.25 and P(spade) = 0.25. A card cannot be both, so P(heart or spade) = 0.5.
At least once. If A has a 30% chance and you get three independent tries, P(at least once) = 1 − 0.7³ = 0.657.
Avoiding common mistakes
Adding without subtracting the overlap
Adding 30% and 40% to get 70% for “rain on at least one day” double-counts the weekends when it rains on both days. Only mutually exclusive events can simply be added.
Assuming independence
Multiplying probabilities is only valid when one event does not change the other’s chances. Weather on consecutive days, defects from the same machine and test answers from the same student are usually correlated. If you know the joint probability, enter it directly.
Impossible combinations
Not every set of numbers is consistent. P(A and B) cannot exceed the smaller of P(A) and P(B), and P(A) + P(B) − P(A and B) cannot exceed 1. The calculator explains which constraint fails.
Related tools
For a fixed number of repeated trials and exact counts of successes, use the binomial probability calculator. Venn diagrams of actual elements, rather than probabilities, are drawn by the set operations calculator. To weigh outcomes by payoff, try the expected value calculator, and to convert betting odds to probabilities, the odds calculator.
Frequently asked questions
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other's chances, so both can happen and P(A and B) = P(A) × P(B). Mutually exclusive events cannot happen together, so P(A and B) = 0. Two events with nonzero probabilities can never be both: if they exclude each other, knowing one happened tells you the other did not, which is dependence.
When do I add probabilities and when do I multiply?
Add for or, multiply for and, with corrections. P(A or B) = P(A) + P(B) − P(A and B); the subtraction removes the overlap counted twice. P(A and B) = P(A) × P(B | A), which simplifies to P(A) × P(B) only when the events are independent.
How do I find the probability of at least one occurrence?
Use the complement. The chance an event with probability p never happens in n independent tries is (1 − p)^n, so the chance it happens at least once is 1 − (1 − p)^n. A 30% event tried three times happens at least once with probability 1 − 0.7³ = 0.657.
What does P(A | B) mean?
It is the probability of A given that B has happened, calculated as P(A and B) ÷ P(B). It restricts attention to the outcomes where B is true. P(A | B) and P(B | A) are usually different; confusing them is a classic error in medical testing and courtrooms.
Can I enter percentages?
Yes. Switch the input format to percentages and type 30 for 30%. Results are shown as decimals with percentages alongside the headline values.