Tree Diagrams with Conditional Probability (Edexcel IAL Maths)

Further Tree Diagrams

Spec point — 3.4

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, $P ( B \vert A )$ 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, $\mathrm{P} ( B \vert A ' )$ would require starting with event “$notA$ ” in the 1^stexperiment 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

• $P ( B \vert A )$ (“given that”) is the probability on the branch of the 2^nd experiment

• However, the “given that” statement $P ( A \vert B )$ is more complicated and a matter of working backwards

  • from Conditional Probability, $\mathrm{P} ( A \vert B ) = \frac{ \mathrm{P} ( A \cap B )}{ \mathrm{P} ( B )}$
  • from the diagram above, $P(B)=P(A\cap B)+P(A'\cap B)$
  • leading to $P ( A \vert B ) = \frac{P ( A \cap B )}{P ( A \cap B ) + P ( A ' \cap B )}$
  • This is quite a complicated looking formula to try to remember so use the logical steps instead – and a clearly labelled tree diagram!

Worked Example The event $F$ has a 75% probability of occurring.

The event $W$ follows event $F$, and if event $F$ has occurred, event $W$ has an 80% chance of occurring.

It is also known that $P(F'\cap W)=0.15$ .

Find

(i) $\mathrm{P} ( W \vert F ' )$

(ii) $\mathrm{P} ( F \vert W ' )$

(iii) the probability that event $F$ didn’t occur, given that event $W$didn’t occur.

Answer:

Exam Hint

• It can be tricky to get a tree diagram looking neat and clear first attempt – it can be worth drawing a rough one first, especially if there are more than two outcomes or more than two events; do keep an eye on the exam clock though!

• Always worth another mention – tree diagrams make particularly frequent use of the result $P(\mathrm{not}A)=1-P(A)$

• Tree diagrams have built-in checks

  • the probabilities for each pair of branches should add up to 1
  • the probabilities for each outcome of combined events should add up to 1