How it works

For independent events, the probability of both happening is found by multiplying their individual probabilities. Probability trees display the outcomes of multiple events, with branches multiplied along each path and probabilities of separate paths added together. For dependent events (without replacement), probabilities on later branches must be adjusted since the sample space changes.

The method

1. 1.Draw a probability tree diagram showing all outcomes
2. 2.Label each branch with its probability
3. 3.Multiply along branches to find the probability of a combined outcome
4. 4.Add probabilities of different branches that satisfy the event

Worked example

A bag has 3 red and 2 blue balls. Find P(two reds) with replacement.

1. P(red) = 3/5 each time
2. P(two reds) = 3/5 × 3/5 = 9/25

Worked example

Find P(two reds) without replacement from the same bag.

1. First red: 3/5. Second red: 2/4
2. P = 3/5 × 2/4 = 6/20 = 3/10

Common mistakes

• Adding probabilities along a single branch instead of multiplying
• Not adjusting probabilities for events without replacement
• Forgetting to add the probabilities of separate valid outcome paths