Probability tree diagrams

In a nutshell

Tree diagrams are used in probability to help visually illustrate the probability of certain events occurring. They are comprised of nodes which represent the different outcomes of an experiment and branches showing the probabilities of each outcome.

Constructing a tree diagram

Procedure

1.	Draw the starting node.
2.	Plot all possible outcomes of the first experiment as nodes.
3.	Write down the probabilities of each outcome as branches from the starting node.
4.	Repeat the process, treating each new node as a starting node.

Note: Watch out for the phrase 'without (or with) replacement' as this tells you whether the probabilities will change across nodes.

Example 1

Max has a bag containing red and green balls. There are seven red balls and three green balls in the bag. If he draws two balls without replacement, construct a tree diagram illustrating the different possible outcomes.

Calculate the total number of balls in the bag.

$7+3=10$

Calculate the probability of drawing each ball on the first go.

$\begin{aligned}P(\text{Red)}=\dfrac{7}{10}\\P(\text{Green)}=\dfrac{3}{10}\end{aligned}$

Calculate the probabilities after a green ball is picked.

$\begin{aligned}P(\text{Red)}=\dfrac{7}{10-1}=\dfrac{7}{9}\\P(\text{Green)}=\dfrac{3-1}{10-1}=\dfrac{2}{9}\end{aligned}$

Calculate the probabilities after a red ball is picked.

$\begin{aligned}P(\text{Red)}=\dfrac{7-1}{10-1}=\dfrac{6}{9}\\P(\text{Green)}=\dfrac{3}{10-1}=\dfrac{3}{9}\end{aligned}$

Draw a tree diagram like so:

Maths; Probability; KS4 Year 10; Probability tree diagrams

Note: You will also see tree diagrams drawn horizontally.

Finding probabilities from tree diagrams

To obtain the probability of an event from a tree diagram, multiply the probabilities across (or down) the branches, connecting the starting node to the node of the event.

Example 2

Using the tree diagram below, find the probability of choosing a red ball and then a green ball.

Maths; Probability; KS4 Year 10; Probability tree diagrams

Multiply down the red branch and then the green branch.

$P(\text{Red, Green})=\dfrac{7}{10} \times \dfrac{3}{9} = \underline{\dfrac{21}{90}}$