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Find an event probability from a tree diagram

Multiply along paths, add across paths

The probability of one complete path through a tree is the product of its branch probabilities. If an event can occur on several mutually exclusive paths, add those path probabilities.

Problem

A tree starts with $P (A) = 0.25$ and $P (not A) = 0.75$ . After A, the chance of B is 0.2; after not A, the chance of B is 0.6. What is the overall probability of B?

Answer

P (B) = 0.5

The two paths to B are disjoint and cover every way B can occur.

Step-by-step solution

1. Follow the path through A

The probability of both A and B is $0.25 (0.2) = 0.05$ .

2. Follow the path through not A

The probability of not A followed by B is $0.75 (0.6) = 0.45$ .

3. Add the disjoint paths

Either path produces B, so $P (B) = 0.05 + 0.45 = 0.50$ .

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