For independent events, the AND rule multiplies: P(A and B) = P(A) × P(B). For any two events, the OR rule adds and then subtracts the overlap: P(A or B) = P(A) + P(B) − P(A and B). If the events are mutually exclusive, the overlap is 0 and the OR rule becomes a simple addition.
AND and OR rules
- AND (independent events): P(A and B) = P(A) × P(B)
- OR (any events): P(A or B) = P(A) + P(B) − P(A and B)
- OR (mutually exclusive): P(A or B) = P(A) + P(B)
- Independent if P(A and B) = P(A) × P(B)
Independent events and the AND rule
Events are independent if the outcome of one does not affect the other, such as two spins of a spinner. For independent events, multiply to find the probability that both happen. If P(red) = 0.45 on each spin, P(red on both spins) = 0.45 × 0.45 = 0.2025. The rule extends to three or more events.
The OR rule with an overlap
When two events can happen together, adding P(A) and P(B) counts the overlap twice. Subtract it once: P(A or B) = P(A) + P(B) − P(A and B). With P(A) = 0.55, P(B) = 0.4 and P(A and B) = 0.25, P(A or B) = 0.7.
Mutually exclusive or independent?
These are different ideas. Mutually exclusive events cannot happen together, so P(A and B) = 0. Independent events can happen together, but one does not affect the other. Two events with non-zero probabilities cannot be both mutually exclusive and independent.
Testing independence
To check whether two events are independent, compare P(A and B) with P(A) × P(B). With P(A) = 0.55 and P(B) = 0.4, the product is 0.22, but P(A and B) = 0.25, so the events are not independent.
Combining the rules
For independent events, find P(A and B) by multiplying first, then use it in the OR rule. With P(A) = 0.35 and P(B) = 0.65, P(A and B) = 0.2275 and P(A or B) = 0.35 + 0.65 − 0.2275 = 0.7725.
Worked examples
Example 1. A spinner lands on red with probability 0.45. It is spun twice. Find P(red on both spins).
- The spins are independent: 0.45 × 0.45.
Answer: 0.2025.
Example 2. A and B are independent, with P(A) = 0.35 and P(B) = 0.65. Find P(A and B) and P(A or B).
- P(A and B) = 0.35 × 0.65 = 0.2275.
- P(A or B) = 0.35 + 0.65 − 0.2275.
Answer: P(A and B) = 0.2275; P(A or B) = 0.7725.
Example 3. P(A) = 0.55, P(B) = 0.4 and P(A and B) = 0.25. Find P(A or B).
- 0.55 + 0.4 − 0.25.
Answer: 0.7.
Example 4. For the events in the previous example, are A and B independent?
- P(A) × P(B) = 0.55 × 0.4 = 0.22.
- P(A and B) = 0.25, which is different.
Answer: No, because P(A and B) is not equal to P(A) × P(B).
Example 5. A student passes three independent tests with probabilities 0.9, 0.85 and 0.7. Find P(passes all three).
- Multiply: 0.9 × 0.85 × 0.7.
Answer: 0.5355.
Common mistakes
- Adding for AND. For independent events, P(A and B) = P(A) × P(B).
- Forgetting to subtract the overlap. P(A or B) = P(A) + P(B) − P(A and B).
- Confusing independent with mutually exclusive. Mutually exclusive events cannot happen together; independent events can.
- Multiplying dependent events with the original probabilities. Only multiply unchanged probabilities when the events are independent.
- Getting a probability over 1. If an OR answer is over 1, the overlap has not been subtracted.
How the AND and OR rules are examined
Higher papers use the AND and OR rules on their own and inside almost every tree diagram question. Questions ask you to:
- find the probability of independent events both happening
- use the OR rule, including the overlap
- decide whether two events are independent
- combine the rules for three or more events
The AND and OR rules worksheet includes worked answers, and the GCSE probability revision engine provides unlimited practice.
These rules are used in tree diagrams. For events that cannot happen together, see mutually exclusive events.
Frequently asked questions
What is the AND rule in probability?
For independent events, P(A and B) = P(A) × P(B).
What is the OR rule in probability?
P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, the last term is 0.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other; mutually exclusive events cannot happen at the same time.