Tree Diagrams with Conditional Probability (Cambridge International AS Mathematics)

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, $P ( B \vert A )$ then event A has already occurred so start by looking for the branch event A in the 1^st experiment, and then P(B | A) 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 Probability Formulae

• Error converting from MathML to accessible text. (“given that”) is the probability on the branch of the 2^nd experiment

• However, the “given that” statement Error converting from MathML to accessible text. 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

• 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!