Combined events and conditional probability · GCSE Statistics

Mutually exclusive and independent events

Mutually exclusive events can't happen at the same time, such as rolling a 2 and a 5 on one dice. P(A or B) = P(A) + P(B).
Independent events don't affect each other, such as flipping a coin and rolling a dice. P(A and B) = P(A) × P(B).
For events that are not mutually exclusive, P(A or B) = P(A) + P(B) − P(A and B). With P(A) = 0.4, P(B) = 0.5 and P(A and B) = 0.2, P(A or B) = 0.7.

Tree diagrams show combined events. Each set of branches adds up to 1.
Multiply along the branches to find the probability of a route. Add the probabilities of routes that give the outcome you want.
With replacement: the probabilities stay the same. A bag has 3 red and 2 blue balls; P(red, red) = 3/5 × 3/5 = 9/25.
Without replacement: the second probabilities change. P(red, red) = 3/5 × 2/4 = 6/20 = 3/10.

Venn diagrams show events as overlapping circles. The overlap is "A and B", and the region outside the circles is "neither".
Conditional probability is the probability of B given that A has happened: P(B given A) = P(A and B) ÷ P(A).
Example: 30 students play sport and 12 of them play piano. P(plays piano given plays sport) = 12 ÷ 30 = 0.4.
If P(B given A) = P(B), the events are independent.

Mutually exclusive events: Events that can't happen at the same time.
Independent events: Events where one doesn't affect the other.
Tree diagram: A diagram showing the outcomes of combined events on branches.
Conditional probability: The probability of an event given that another has happened.
Venn diagram: A diagram of overlapping circles showing events.
Without replacement: When an item is not put back, so probabilities change.