3.2.3 Further Tree Diagrams

The tree diagrams used here are no more complicated than those in the first Tree Diagrams revision note, however
- The wording/terminology used in questions and on diagrams may now involve the use of set notation including the symbols $\cup$ (union), $\cap$ (intersection) and ‘ (complement)
  - e.g. P(A') would be used for “P(not A)”
- Conditional probability questions can be solved 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, $P (B | A)$ then event A has already occurred so start by looking for the branch event A in the 1^st experiment, and then would be the branch for event in the 2^nd experiment

Similarly, $P (B | A)$ would require starting with event “ $n o t A$ ” 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
$P (B | A)$ (“given that”) is the probability on the branch of the 2^nd experiment
However, the “given that” statement $P (A | B)$ is more complicated and a matter of working backwards
- from Conditional Probability, $P (A | B) = \frac{P (A \cap B)}{P (B)}$
- from the diagram above, $P (B) = P (A \cap B) + P (A' \cap B)$
- leading to $P (A | B) = \frac{P (A \cap B)}{P (A \cap B) + P (A' \cap B)}$
- This is quite a complicated looking formula to try to remember so use the logical steps instead – and a clearly labelled tree diagram!

The event 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)

P (W | F')

(ii)

P (F | W')

(iii)

the probability that event

F

didn’t occur, given that event

W

didn’t occur.

3-2-3-fig2-we-solution-part-1

2ACeFam__3-2-3-fig2-we-solution-part-2

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 $P (not A) = 1 - P (A)$
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

AQA A Level Maths: Statistics

3.2.3 Further Tree Diagrams