Probability tree diagrams

Years 8 to 13

A tree diagram shows every outcome of two events and how to multiply along the branches.

Two picks from 3 red and 2 blue counters, without replacement. The chance of red then red is 3/5 × 2/4.

For learners

A tree diagram shows what can happen, one step at a time. Each branch has a chance written on it.

Here is a bag with 3 red and 2 blue counters. Pick one, then pick another without putting the first back. The chances for the second pick change, because there are fewer counters.

To find the chance of a whole path, multiply along the branches. Red then red: $\frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$ .

A worked example

A bag has 3 red and 2 blue counters. Two are taken without replacement. Find P(both red).

First red: $\frac{3}{5}$ .
Second red (2 reds left out of 4): $\frac{2}{4}$ .
$\frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$ .

For grown-ups

Probabilities on the branches leaving one point must add to 1. Multiply along a path to get its probability, and add the paths when more than one path gives the result you want.

Common mistakes

Not changing the second set of probabilities when nothing is put back.
Adding along a path instead of multiplying.
Forgetting the second order: 'different colours' needs red-then-blue and blue-then-red.

Why it matters

Probability trees are used in games, medical tests, weather forecasts and risk.

Topics in quarter that use this

Probability of two events
Conditional probability
Simple probability
Listing outcomes
The binomial distribution