Probability calculator (AND, OR and at least one)
Enter the probability of each event and this calculator produces every combined probability an exam question can ask for — both events, either event, neither event, exactly one, and at least one — as decimals, fractions of 1 and percentages, with the multiplication and addition rules written out.
Enter two probabilities between 0 and 1
P(A and B) = 0.18
P(A or B): 0.72 · P(neither): 0.28
P(exactly one): 0.54 · P(at least one): 0.72
P(not A): 0.4 · P(not B): 0.7
Both events happen 18% of the time.
Working
- P(not A) = 1 − 0.6 = 0.4 and P(not B) = 1 − 0.3 = 0.7
- P(A and B) = 0.6 × 0.3 = 0.18 — multiply along the branches.
- P(A or B) = 0.6 + 0.3 − 0.18 = 0.72 — subtract the overlap.
- P(neither) = 0.4 × 0.7 = 0.28
- P(exactly one) = 0.6×0.7 + 0.4×0.3 = 0.54
- P(at least one) = 1 − P(neither) = 0.72
The method behind it
- Multiply along branches, add between branches.
- P(A and B) = P(A) x P(B) for independent events.
- P(A or B) = P(A) + P(B) - P(A and B).
- P(at least one) = 1 - P(neither).
How to use this tool
1. Enter each probability
Type P(A) and P(B) as decimals between 0 and 1. A percentage such as 30% is 0.3, and a fraction such as 3/10 is also 0.3.
2. Choose independent or given
Independent events do not affect each other, so P(A and B) = P(A) x P(B). If the second probability changes after the first event, enter that changed value as P(B).
3. Read the branch working
The steps mirror a probability tree: multiply along branches, then add the branch results that satisfy the outcome you want.
Frequently asked questions
How do you find the probability of A and B?
Multiply along the branches: for independent events P(A and B) = P(A) x P(B). If the events are dependent, use the conditional probability of B after A has happened.
How do you find the probability of A or B?
Use the addition rule P(A or B) = P(A) + P(B) - P(A and B). Subtracting the overlap stops you counting the outcome where both happen twice.
What is the quickest way to find at least one?
Use the complement: P(at least one) = 1 - P(neither) = 1 - (1 - P(A))(1 - P(B)). It is one calculation instead of adding three branches.
Do probabilities have to add up to 1?
The probabilities of all the possible outcomes of one trial add to 1, so P(not A) = 1 - P(A). Two different events do not have to add to 1 unless they are the only outcomes.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.