Tree diagrams

Tree diagrams are a way of showing combinations of two or more events. Each branch is labelled at the end with its outcome and the probability is written alongside the line.

Two events are independent if the probability of the first event happening has no impact on the probability of the second event happening.

For example, the probability of rolling a 6 on a die will not affect the probability of rolling a 6 the next time. The scores on each roll are independent.

If a die was to be rolled twice, the tree diagram would look like this:

Tree diagram showing different options of getting a 6 on each dice

There are four possible outcomes. To work out the probabilities of each combination, multiply the probabilities together.

Fractional probabilities of getting a 6 on each dice
Question

A bag only contains 4 blue counters and 3 red counters. A box only contains 5 blue counters and 2 red counters. One counter is taken at random from the bag and one counter is taken at random from the box. Complete the tree diagram and work out the probability of selecting two red counters.

Tree diagram showing probability of getting blue and blue in both bag and box

Use the fact that probabilities add up to 1 to work out the probabilities of the missing branches.

Tree diagram showing probabilities of getting blue and then a different colour

The probability of selecting two red counters is \frac{3}{7} \times \frac{2}{7} = \frac{6}{49}.