2024 TJC H1 Q9

In a private housing estate, the adult residents are asked to vote for an upgrading of the gym facilities. During the voting, \(\frac{3}{4}\) of the female adult residents and \(\frac{3}{8}\) of the male adult residents cast their votes. Given that \(60\)% of the adult residents are female, show that the probability that an adult resident casts a vote is \(\frac{3}{5}\).[2]

Three adult residents in the estate are randomly chosen after the voting.

1. Find the probability that exactly one of the \(3\) residents casts a vote.[2]
2. Given that all three residents cast a vote, find the probability that exactly two are male.[3]