2024 TJC P2 Q7

Two boys, Joseph and Elliot, play a game by each tossing a coin. Joseph tosses a \(20\)-cent coin and Elliot tosses a \(50\)-cent coin.

The probability that the \(20\)-cent coin and the \(50\)-cent coin show a head are \(\frac{3}{5}\) and \(p\) respectively.

If both coins show heads, Joseph gets to keep Elliot's coin.

If both coins show tails, Joseph gives his coin to Elliot.

If one coin shows a head and the other coin shows a tail, both get to keep their own coins.

Let \(W\), in cents, be the amount of money Joseph wins in a game.

1. Find, in terms of \(p\), the probability distribution of \(W\) and \(\mathrm{E}(W)\).[4]
2. Find the value of \(p\) for the game to be fair.[1]
3. Suppose Elliot's coin is fair i.e. \(p = \frac{1}{2}\) and the boys played \(40\) games. Find the probability that Joseph wins an average of more than \(15\) cents per game.[3]