2023 CJC P2 Q5

During the Great Singapore Sale, a certain electronics store organises a lucky draw to attract more customers. The lucky draw is designed as follows:

A circular board is divided into four sectors labelled with numbers \(1\), \(2\), \(3\), \(4\) and has angles \({{144}^{{}^\circ }}\), \({{108}^{{}^\circ }}\), \({{72}^{{}^\circ }}\), \({{36}^{{}^\circ }}\) respectively. The board has a spinner pivoted at the centre of the circular board. When a customer spins the spinner, the spinner comes to rest randomly in one of the four sectors.

Every customer who visits the store is allowed to play one round of the lucky draw. In each round of the lucky draw, the customer gets to spin the spinner twice.

The score, \(X\), of the customer is

• the sum of the two numbers if the numbers from the two spins are different,

• three times the number if the numbers from the two spins are the same.

A customer wins a prize if the score, \(X\), is more than \(6\).

1. Show that \(\mathrm{P}\left( X=6 \right)=0.15\).[2]
2. Find the probability distribution of \(X\).[3]
3. Find the probability that a customer scores less than \(10\), given that the customer wins a prize.[3]