2019 ACJC CT Q4

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 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 \(P\left( X=10 \right)=0.335\).[1]
2. Tabulate the probability distribution of \(X\).[2]
3. Find the expectation of the total value of cash vouchers won.[2]

Let \(D\) represent the number of \(\$10\) cash vouchers won and \(S\) represent the number of a \(\$5\) cash vouchers. Use your answer in part (iii) to find \(E\left( D \right)\)and \(E\left( S \right)\).[2]

Explain why the same method cannot be used to find \(\mathrm{Var}\left( D \right)\) and \(\mathrm{Var}\left( S \right)\).[1]