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Nov 2009 p62 q3
2638
Maria chooses toast for her breakfast with probability 0.85. If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8. If she has a bread roll then the probability that she will have jam on it is 0.4.
(i) Draw a fully labelled tree diagram to show this information.
(ii) Given that Maria did not have jam for breakfast, find the probability that she had toast.
Solution
(i) The tree diagram is structured as follows:
First branch: Toast (T) with probability 0.85, Bread roll (B) with probability 0.15.
Second branch from Toast: Jam (J) with probability 0.8, No Jam (NJ) with probability 0.2.
Second branch from Bread roll: Jam (J) with probability 0.4, No Jam (NJ) with probability 0.6.
(ii) To find the probability that Maria had toast given she did not have jam, use Bayes' theorem:
\(P(T \mid NJ) = \frac{P(T \text{ and } NJ)}{P(NJ)}\)