2023 TJC P2 Q5

A group of \(12\) artistes, comprises \(7\) men and \(5\) women are invited to participate in an episode of TV variety show. In one of the challenges, \(3\) pairs of artistes, each consisting of a man and a woman, are to be selected to play a game in a night market.

1. Find the number of ways the \(3\) pairs can be selected and grouped into pair \(A\), \(B\) and \(C\).[2]

In the game, the woman is required to generate a number, \(1\) to \(9\), by using a number generator with the probability of obtaining a number \(x\) is \(\frac{1}{45}x\).

If the number is an even number, the male artiste must draw a ball from a red box with \(3\) black and \(5\) yellow balls.

If the number is odd, he must draw a ball from a blue box with \(8\) black and \(4\) yellow balls.

If a yellow ball is picked, the pair will win. Otherwise, the pair will lose.

2. Pair \(A\) plays the game once. Show that the probability that the pair loses the game is \(\frac{29}{54}\).[2]

Pair \(B\) plays a match with Pair \(C\). Each pair takes turns to play the game.

The ball is replaced before the next pair plays the game.

The match will stop only when either one of them wins a game. Pair \(C\) starts first.

3. Find the probability that Pair \(B\) is the winner.[3]