2015 VJC P2 Q7
A game is to be played between two players, $A$ and $B$. A bag contains $5$ balls each with $A$’s name and $8$ balls each with $B$’s name. Starting from $A$, the players will take turn to pick $3$ balls randomly in a single draw from the bag, note down the names and return all $3$ balls to the bag. Assuming that the balls are identical in size, the winner is the first player to get $3$ balls of the player’s name in a single draw.
(i)
Find the probability that $A$ wins the game.
[4]
(ii)
Find the probability that $A$ wins on $A$’s first draw given that $A$ wins the game.
[3]
Suppose $A$ eventually wins the game on $A$’s $4$th turn. Let (${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$, ${{a}_{4}}$) denote $A$’s draw sequence where ${{a}_{i}}$ denote the number of balls with $A$’s name picked by $A$ on $A$’s $i$th turn.
Find the total number of possible draw sequence for $A$.
[2]
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