2021 DHS P2 Q7

Mike plays a game at a fun-fair. Each game costs \(\$3\) to play. The probability distribution of the amount of money \(\$X\), that Mike wins from a game is given by

\({\mathrm{P}(X=x)=\begin{cases} kx & \text{for}\hspace{0.5em} x=1,2,3, \\ \frac{1}{3}k(x-1) & \text{for}\hspace{0.5em} x=4,5,6, \\ 0 & \textbf{ \quad\: otherwise,} \end{cases}}\)

where \(k\) is a constant.

1. Show that \(k=\frac{1}{10}\).[1]
2. Find the expectation and variance of his net gain for one game.[3]
3. Assuming that each game is independent of one another, find the probability that his total net gain in \(30\) games is more than \(\$20\).[3]