2020 RI P2 Q11

In conjunction with the Great Singapore Sale, a certain electronics store is having a lucky draw for their customers. In each round of the lucky draw, the customer draws two balls randomly, one after another, with replacement from a box containing \(20\) red balls, \(30\) blue balls and \(50\) white balls. The colour of each ball drawn is noted and points are awarded accordingly as follows.

The customer’s score in each round of the lucky draw is the total number of points awarded for the balls drawn. To illustrate, a customer scores a total of \(5\) points if he draws a blue ball and a white ball in a round of the lucky draw, regardless of the order of appearance of the balls.

You may assume that the \(100\) balls are indistinguishable from each other apart from their colour.

Let \(X\) denote the total number of points scored by a customer in a round of the lucky draw.

1. Tabulate the probability distribution of \(X\).[3]
2. Find \(\mathrm{E}\left( X \right)\) and show that \(\mathrm{Var}\left( X \right)=6.02\).[2]

Mr Lim participated in \(50\) rounds of the lucky draw.

3. Using a suitable approximation, find the probability that Mr Lim’s average score is at least 6.[3]

A customer wins a cash voucher if his total score for one round of the lucky draw is more than \(6\).

The total number of cash vouchers won by Mr Tan in \(n\) rounds of the lucky draw is denoted by \(Y\).

4. Find the least value of \(n\) such that there is a probability of more than \(0.7\) that Mr Tan will win more than \(3\) cash vouchers in total.[4]