Tree diagrams for multi-stage experiments

By the end of this lesson

By the end you can build a tree diagram for a two-stage experiment, use the rule multiply along, add across to get probabilities, and deal correctly with sampling without replacement.

Why a picture helps

A tree diagram organises a multi-stage experiment into stages. Each stage branches into its possible results, and each branch is labelled with its probability.

A path is a route from the starting point on the left, following one branch at each stage.
A path ends at a leaf: one of the end points on the far right.
Each path stands for one complete outcome, such as "red, then blue".

Two rules do all the work:

Multiply along a path to get the probability of that whole path.
Add across the paths that make up an event.

Once the tree is drawn, most of the thinking is already finished.

Compare the second-stage fractions: they differ depending on the first draw, because the bag has changed. The four path probabilities add to

1

, which is a useful check that the tree is complete.

Worked example: Example 1: both red, and both blue

A bag holds $4$ red and $6$ blue counters. Two counters are taken at random without replacement. Find the probability that both are red.

Have a go first?

Check 1 — multiplying along

CalculateQuestion 1 · Apply

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.