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June 2018 p63 q3
2601
The members of a swimming club are classified either as ‘Advanced swimmers’ or ‘Beginners’. The proportion of members who are male is \(x\), and the proportion of males who are Beginners is 0.7. The proportion of females who are Advanced swimmers is 0.55. This information is shown in the tree diagram.
For a randomly chosen member, the probability of being an Advanced swimmer is the same as the probability of being a Beginner.
(i) Find \(x\).
(ii) Given that a randomly chosen member is an Advanced swimmer, find the probability that the member is male.
Solution
(i) Let the probability of being a Beginner be 0.5. The probability of being a Beginner is given by:
\(0.7x + 0.45(1-x) = 0.5\)
Solving for \(x\):
\(0.7x + 0.45 - 0.45x = 0.5\)
\(0.25x = 0.05\)
\(x = 0.2\)
(ii) The probability of being an Advanced swimmer is also 0.5. The probability of being an Advanced swimmer and male is:
\(0.3x = 0.3 \times 0.2 = 0.06\)
The probability of being an Advanced swimmer is:
\(0.5 = 0.3 \times 0.2 + 0.55 \times 0.8\)
The probability that a randomly chosen Advanced swimmer is male is: