2016 NJC P2 Q10

1. Suppose a fair die is tossed twice. Calculate the probabilities that
   1. the sum of the scores of the two tosses is at least \(8\),[1]
   2. the absolute difference between the scores of the two tosses is at least \(4\).[1]

In one round of a game, a player is to draw a ball, without replacement, from a box that contains \(3\) red balls and \(4\) white balls. If a red ball is drawn, the player will add the scores obtained from tossing a fair die twice. If a white ball is drawn, the player will take the absolute difference of the scores obtained from tossing a die twice.

The game ends if the sum of the scores is at least 8 or the absolute difference of the scores is at least \(4\). Else, the player will proceed to the second round of the game where the process of picking a ball from the box and tossing the die twice repeats.

2. Find the probability that the game ends at the first round.[2]
3. Suppose the game ends at the first round. Find the probability that a red ball is drawn.[2]
4. Find the probability that there are a total of \(3\) rounds of game played and exactly \(2\) white balls are selected.[3]