2025 ASRJC P2 Q7

Anthony plays a game using a four-sided fair die with the numbers 1, 2, 3 and 4 printed on its sides. Anthony rolls the die and should the outcome be a prime number, he wins an amount equivalent to that number in dollars. Conversely, if the outcome is a non-prime number, Anthony will lose an amount equivalent to that number in dollars. The random variable \(X\) represents the amount of money Anthony wins in a roll of the die.

1. Tabulate the probability distribution of \(X\) and find the value of \(E(X)\).[2]
2. Find the value of the variance of \(X\).[1]
3. The random variables \(X_1\) and \(X_2\) are two independent observations of \(X\). By drawing a table of outcomes for \(|X_1 + X_2|\) or otherwise, find \(E(|X_1 + X_2|)\).[2]

Assume now that a biased six-sided die with numbers 2, 3, 4, 5, 6 and 7 printed on its sides, is used instead for the same game. It is known that for this biased die, the probability of obtaining a "3" is \(p\), the probability of obtaining a "5" is \(2p\) and the remaining four numbers each have an equal probability of occurring.

4. Find the exact value of \(p\) for the game to be fair using the biased die.[3]