2019 CJC P2 Q6

To raise its profile, ABC Supermart devised a publicity where, if a customer has purchases of \(\${}R\) or more in a single receipt, he is qualified to participate in a sure-win “Lucky Spin” game. In the game, the customer spins the first wheel (which is divided into \(3\) equal parts) to see how much money he wins, then he spins a second wheel to see how much his winning is multiplied by.

For example, based on the illustration above, if the result from the first spin is “\(\$10\)”, and the result of the second spin is“\(\times 4\)”, then the person playing the game would have received \(\$40\).

Let \(X\) denote the amount won from playing the game.

1. Tabulate the probability distribution of \(X\).[2]
2. ABC Supermart makes a profit of \(40\)% on all purchases made by customers. It wishes to use the profits generated from every qualifying receipt to offset the amounts given away in the game. Determine the value of \(\${}R\) that the Supermart needs to set for a customer to qualify to play the game. Give your answer correct to the nearest dollar.[3]
3. Find the value of \(\sigma\), the standard deviation of \(X\). Hence find \(\mathrm{P}\left( X<\sigma \right)\).[3]