Tree Diagrams with Conditional Probability (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Revision Note

Further Tree Diagrams

What do you mean by further tree diagrams?

• The tree diagrams used here are no more complicated than those in the first Tree Diagrams revision note, however

  • questions may use set notation as well, or alongside contextual questions $\cup$(union),  $\cap$(intersection) , ‘ (complement), | ("given that")

  • more detailed use of conditional probability

  • three events for each experiment and three experiments could be used

How do I solve conditional probability problems using tree diagrams?

• Interpreting questions in terms of AND ($\cap$), OR ($\cup$), complement ( ‘ ) and “given that” ( | )

• Condition probability may now be involved too

• 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 P(B | A)  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 Probability Formulae

• $\mathbf{P}\mathbf{\left(}\mathit{B}\mathbf{\right|}\mathit{A}\mathbf{)}$ (“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!