2021 RI P2 Q10

A bag contains four balls numbered \(1\),\(2\),\(3\) and \(4\). In a game, a ball is drawn at random from the bag and then a fair coin is tossed a number of times that is equal to the number shown on the ball drawn. The random variable \(X\) is the number of heads recorded.

1. Show that \(P\left( X=0 \right)=\frac{15}{64}\). Find \(\mathrm{P}\left( X=x \right)\) for all other possible values of \(x\).[5]
2. Denoting the expectation and variance of \(X\) by \(\mu \) and \({{\sigma }^{2}}\) respectively, find \(\mathrm{P}\left( X>\mu \right)\) and show that \({{\sigma }^{2}}=\frac{15}{16}\).[3]

Adam plays this game \(10\) times.

3. Find the probability that there are at least two games with at least \(2\) heads recorded.[2]

Bill plays this game \(50\) times.

4. Using a suitable approximation, estimate the probability that the average number of heads recorded is less than \(1\).[3]