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Tree diagrams and combined events
Tree diagrams show two or more events step by step, so you can find combined probabilities.
AND and OR
Independent events do not affect each other.
P(A and B) = P(A) × P(B): multiply along the branches.
P(A or B) for outcomes that cannot happen together: add the results.
P(A and B) = P(A) × P(B): multiply along the branches.
P(A or B) for outcomes that cannot happen together: add the results.
Tree diagrams
Each set of branches adds to 1.
Two coin throws: P(two heads) = 0.5 × 0.5 = 0.25.
P(exactly one head) = (H, T) + (T, H) = 0.25 + 0.25 = 0.5.
Two coin throws: P(two heads) = 0.5 × 0.5 = 0.25.
P(exactly one head) = (H, T) + (T, H) = 0.25 + 0.25 = 0.5.
Without replacement
If the first item is not put back, the second set of branches changes.
3 red and 2 blue, two picked without replacement:
P(red, red) = 35 × 24 = 620 = 310
3 red and 2 blue, two picked without replacement:
P(red, red) = 35 × 24 = 620 = 310
P(pass driving test) = 0.6 each time, independently. Find P(pass on the second attempt only).
- Fail then pass: 0.4 × 0.6
Answer: 0.24
Multiply along branches for AND, add separate paths for OR. Each set of branches adds to 1. Without replacement, the second-stage probabilities change.
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