2022 TGM P2 Q6

The following diagram shows a board which is divided into three regions \(A\), \(B\) and \(C\).

A game consists of a contestant throwing one dart at the board. The probability of hitting each region is given in the following table.

The contestant obtains points as shown in the following table

Let \(X\) be the number of points scored by Alan in a game.

1. Given that the game is fair for Alan, find the value of \(p\).[3]
2. Find the value of \(\mathrm{Var}(X)\).[1]
3. The random variable \(Y\) is defined by \(X_1 + X_2\), where \(X_1\) and \(X_2\) are \(2\) independent observations of \(X\). Find
   1. \(P(Y \le 2)\),[3]
   2. \(P(Y \ge 0 \mid Y \le 2)\).[2]