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June 2012 p61 q2
2631
Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3. On station 1 the probability that the presenter is male is 0.1, on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is \(p\). When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075.
Show that the value of \(p\) is 0.3.
Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.
Solution
(i) The probability that the radio is set to station 3 and the presenter is male is given by:
\(0.25p = 0.075\)
Solving for \(p\):
\(p = \frac{0.075}{0.25} = 0.3\)
(ii) To find the probability that the radio was set to station 2 given that Maria hears a male presenter, we use conditional probability:
\(P(2|M) = \frac{P(2 \text{ and } M)}{P(M)}\)
Where:
\(P(2 \text{ and } M) = 0.45 \times 0.85 = 0.3825\)