2026 EJC P2 Q6

A fair tetrahedral die has its four faces numbered 1, 2, 3 and 4. A game is played by rolling the die twice, and the number on the face touching the table is recorded each time. Let \(X\) be the score obtained from the two rolls, defined as follows:

• If the two numbers recorded are different, \(X\) is the absolute difference between them.
• If the two numbers recorded are the same, \(X\) is their common value.

1. Find the probability distribution of \(X\).[2]
2. Find the expectation and variance of \(X\).[2]

To encourage citizens to adopt an active lifestyle, a town council awards fitness cash vouchers to participants. Each participant plays the game three times and is awarded a \(\$2\) cash voucher for every point scored. The participant cashes out the vouchers only after playing the game three times, where a one-time administration fee of \(\$0.50\) is deducted. Let \(W\) be the random variable denoting the value, in dollars, of the cash vouchers received by a participant after the administration fee has been deducted.

3. Find \(\mathrm E(W)\) and \(\mathrm{Var}(W)\).[3]