2.3.2 Further Tree Diagrams

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

2-3-2-cie-fig0-three-tree

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 P(B | A) would be the branch for event B 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 $F$ 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)

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

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

Be wary of assuming that “given that” statements will always be referring to something on the second set of branches (2^nd experiment), they can work the other way!