2020 TJC P2 Q6

Javier and Kelvin played a game with a pack of ten cards numbered \(1\), \(2\), \(3\), …, \(10\). They take turns at drawing two cards each from the pack at random, one card at a time without replacement, and the two cards drawn are replaced before the next player draws. The player who first gets two cards whose sum is \(10\) wins the game. If he does not win, he replaces the two cards and the other player then takes his turn. The game carries on until a winner is decided.

On their first game, Javier gets to draw first.

1. Show that the probability that Javier wins the game on his first draw is \(\frac{4}{45}\).[2]
2. Find the probability that Kelvin wins the game.[3]

On their second game, they toss a biased coin to decide on the player to draw first. Using this biased coin, Kelvin is twice as likely as Javier to draw first.

3. Find the probability that Javier wins the game.[3]