Conditional probability: Tree diagrams

In a nutshell

Tree diagrams are very useful when finding conditional probabilities.

Tree diagrams

On a tree diagram, the probability on the second branch represents the probability of $B$ happening given that $A$ has/has not occurred:

Maths; Conditional probability; KS5 Year 13; Conditional probability: Tree diagrams

Example 1

Consider an experiment with the following probabilities: $P(N)=\dfrac{1}{3}$ , $P(M|N')=\dfrac{1}{2}$ and $P(M'|N)=\dfrac{2}{3}$ . Find $P(N\cap M)$ .

Draw and complete a tree diagram as follows:

Maths; Conditional probability; KS5 Year 13; Conditional probability: Tree diagrams

Find $P(N\cap M)$ by multiplying the first set of branches:

$\begin{aligned} P(N \cap M) &= P(N) \times P(M|N) \\\\&= \dfrac 1 3 \times \dfrac 1 3 \\\\&= \dfrac 1 9\end{aligned}$ 

So, $\underline{P(N\cap M)=\dfrac{1}{9}}$ .