Of all the Sunny Club members in a particular city, 25% prefer swimming on weekends and 75% prefer swimming on weekdays.

Question

It is found that 10% of the members in that city prefer swimming on weekends and are female, while 55% of the members in that city prefer swimming on weekdays and are female.

The probability that a club member picked randomly is female, given that the person prefers swimming on weekends, is 0.19, 0.20, 0.24, or 0.40.

pick one of the four

Answer 1

Lets say Event A is preferring to swim on a Weekend, thus A compliment (Ac) is preferring to swim on a weekday.

So we know P(A) = 0.25 and P(Ac) = 0.75

Then we are told that P(Event A and being female) = 0.10 and P(Ac and being female) = 0.55

The question is asking P(being female | A) = ?

We can use Bayes formula here P(B|A) = probability of event B occurring given we know Event A will occur. This equals the probability of the intersection of Events B and A divided by the probability of Event A occurring.

Thus:

P(being female | Event A) = P(Event A and being female) / P(A) = 0.10/0.25 = 0.4

Thus the probability of a random member being selected when it is known that they prefer swimming on weekends is 0.40 or 40%.