Total Value of Cash Voucher

Two coins are to be flipped. The first coin will land on heads with probability \({0.6}\) and the second coin will land on heads with probability \(0.7\). Assume that the results of the flips are independent. Every head obtained results in a prize, which is equally likely to be either a \({\$10}\) cash voucher or a \({\$5}\) cash voucher.

Let \(X\) represent the total value of cash vouchers won.

1. Show that \(\mathrm{P}\left( X=10\right)=0.335.\)[1]
2. Tabulate the probability distribution of \(X.\)[2]
3. Find the expectation of the total value of cash vouchers won.[2]
4. Let \(D\) represent the number of \(\$10\) cash vouchers won and \(\$5\) represent the number of \(\$5\) cash vouchers. Use your answer in part (iii) to find \({\mathrm{E}\left( D \right)}\) and \(\mathrm{E}\left( S \right).\)[2]
5. Explain why the same method cannot be used to find \(\mathrm{Var}\left( D \right)\) and \(\mathrm{Var}\left( S \right).\)[1]