1

Nov 2009 p62 q3

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.

i Let T, B and X represent toast, bread and not jam, respectively

ii
\[ \begin{array}{l} \mathrm{P}(\mathrm{X})=\mathrm{P}(\mathrm{~T} \text { and } \mathrm{X})+\mathrm{P}(\mathrm{~B} \text { and } \mathrm{X})=(0.85 \times 0.2)+(0.15 \times 0.6) \\ \mathrm{P}(\mathrm{~T} \mid \mathrm{X})=\frac{\mathrm{P}(\mathrm{X} \text { and } \mathrm{T})}{\mathrm{P}(\mathrm{X})}=\frac{0.85 \times 0.2}{(0.85 \times 0.2)+(0.15 \times 0.6)}=\frac{17}{26} \text { or } 0.654 \end{array} \]
The expressions \( \mathrm{P}(\mathrm{T} \) and X\( ) \) and \( \mathrm{P}(\mathrm{X} \) and T\( ) \) have identical meanings and values and are, therefore, completely interchangeable.

2

Nov 2002 p6 q5

Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,
(i) find the conditional probability that Rachel wins the first game, given that she loses the second,
(ii) find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.

3

June 2003 p6 q6

The people living in 3 houses are classified as children \( (C) \), parents \( (P) \) or grandparents \( (G) \). The numbers living in each house are shown in the table below. \begin{array}{|c|c|c|} \hline House number 1 & House number 2 & House number 3 \\ \hline 4 C, 1 P, 2 G & 2 C, 2 P, 3 G & 1 C, 1 G \\ \hline \end{array} (i) All the people in all 3 houses meet for a party. One person at the party is chosen at random. Calculate the probability of choosing a grandparent.
(ii) A house is chosen at random. Then a person in that house is chosen at random. Using a tree diagram, or otherwise, calculate the probability that the person chosen is a grandparent.
(iii) Given that the person chosen by the method in part (ii) is a grandparent, calculate the probability that there is also a parent living in the house.

4

Nov 2003 p6 q5

In a certain country \( 54 \% \) of the population is male. It is known that \( 5 \% \) of the males are colour-blind and \( 2 \% \) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.

5

June 2004 p6 q6

When Don plays tennis, \( 65 \% \) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \( 90 \% \). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \( 80 \% \) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \( 60 \% \). If neither serve goes into the correct area, Don loses the point.
(i) Draw a tree diagram to represent this information.
(ii) Using your tree diagram, find the probability that Don loses the point.
(iii) Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.

6

Nov 2004 p6 q3

When Andrea needs a taxi, she rings one of three taxi companies, \( A, B \) or \( C .50 \% \) of her calls are to taxi company \( A, 30 \% \) to \( B \) and \( 20 \% \) to \( C \). A taxi from company \( A \) arrives late \( 4 \% \) of the time, a taxi from company \( B \) arrives late \( 6 \% \) of the time and a taxi from company \( C \) arrives late \( 17 \% \) of the time.
(i) Find the probability that, when Andrea rings for a taxi, it arrives late.
(ii) Given that Andrea's taxi arrives late, find the conditional probability that she rang company \( B \).

7

June 2007 p6 q2

Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.
(i) Find the probability that Jamie is chosen for the team.
(ii) Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.