Tree Diagrams with Conditional Probability (Edexcel International AS Maths: Statistics 1): Revision Note

Tree diagrams with conditional probability

How do I find conditional probabilities problems from tree diagrams?

• Interpreting questions in terms of AND ($\cap$), OR ($\cup$) and complement ( ‘ )

• Conditional probability may now be involved too - “given that” ( | )

• This makes it harder to know where to start and how to complete the probabilities on a tree diagram

  • e.g. If given, possibly in words, $P\left(B\right|A)$ then event A has already occurred so start by looking for the branch event A in the 1^st experiment, and then there would be the branch for event B  in the 2^nd experiment

• Similarly, $\mathrm{P}\left(B\right|A\text{'})$ would require starting with event “$\mathit{n}\mathit{o}\mathit{t}\mathit{A}$ ” in the 1^st experiment and event B in the 2^nd experiment

• The diagram above gives rise to some probability formulae you will see in the next revision note

• $\mathbf{P}\mathbf{\left(}\mathit{B}\mathbf{\right|}\mathit{A}\mathbf{)}$ (“given that”) is the probability on the branch of the 2^nd experiment

• However, the “given that” statement $\mathbf{P}\mathbf{\left(}\mathit{A}\mathbf{\right|}\mathit{B}\mathbf{)}$ is more complicated and a matter of working backwards

  • from Conditional Probability,  $\mathrm{P}\left(A\right|B)=\frac{\mathrm{P}(A\cap B)}{\mathrm{P}\left(B\right)}$

  • from the diagram above, $P\left(B\right)=P(A\cap B)+P(A\text{'}\cap B)$

  • leading to  $\mathbf{P}\mathbf{\left(}\mathit{A}\mathbf{\right|}\mathit{B}\mathbf{)}\mathbf{=}\frac{\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}}{\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\text{'}}\mathbf{\cap }\mathbf{B}\mathbf{)}}$

  • This is quite a complicated looking formula to try to remember so use the logical steps instead – and a clearly labelled tree diagram!