2018 AJC MYE P2 Q4

In a game, a computer will generate \(3\) distinct random numbers from \(1\) to \(8\). At the same time, a player will draw \(4\) random cards numbered \(1\) to \(8\) without replacement. The numbers drawn by the player will be matched with the numbers generated by the computer.

The number of matching numbers drawn by the player and the computer is denoted by \(X\).

1. Show that \(\mathrm{P}\left( X=1 \right)=\frac{3}{7}\).[1]
2. Find the probability distribution of \(X\).[3]

A player will receive a free movie ticket for \(1\) matching number and a meal coupon for \(2\) matching numbers. If a player gets all matching numbers, he will receive a meal coupon and an additional chance to play the game.

When the player plays the second game, he will win an additional free movie ticket if there is at most \(1\) matching number. An additional meal coupon will be won if the player gets at least \(2\) matching numbers. Each player can only play the game twice.

3. Find the probability that a player wins a free movie ticket if he does not return home empty handed. Show your working clearly.[4]