2025 YIJC P2 Q7

Customers at a shopping mall participate in a game involving \(2\) boxes of tokens, labelled \(A\) and \(B\). Box \(A\) contains \(n\) black tokens and \(4\) gold tokens, while Box \(B\) contains \(4\) black tokens and \(4\) silver tokens. The tokens are identical in all aspects except for their colour.
In a game, a customer first tosses a fair coin. If a head is obtained, two tokens are drawn at random from Box \(B\). Otherwise, one token is drawn from each box. A black token scores \(1\) point, a silver token scores \(2\) points, and a gold token scores \(3\) points. The total number of points is the customer’s score \(X\).

1. Show that \(\mathrm{P}(X=2) = \frac{3}{28} + \frac{n}{4(n+4)}\).[1]
2. Given that \(\mathrm{P}(X=2) = \frac{3}{14}\), determine the probability distribution of \(X\).[4]
3. Find \(\mathrm{P}(X_1=4 \mid X_1 + X_2 \geq 9)\), where \(X_1\) and \(X_2\) are independent observations of \(X\).[3]