2022 HCI P2 Q9

Abel has \(1\) white bag of marbles and \(1\) black bag of cards and he uses them to create a game. The white bag contains \(2\) red marbles, \(x\) blue marbles and \(2x-1\) yellow marbles, where \(x>1\). The black bag has \(3\) cards, and the cards are numbered with the number ‘\(0\)’, ‘\(1\)’ and ‘\(2\)’ respectively.

In each round, Abel will first choose a marble randomly from the white bag. A red marble will give a score of \(4\). If a non-red marble is chosen, Abel will then choose a card randomly from his black bag. The score will then be twice of the number shown on the card drawn.

1. Find the probability that Abel will get a score of \(4\) in a round of the game. Leave your answer in terms of \(x\).[2]
2. Find the probability distribution of Abel’s score in a round of the game in terms of \(x\).[2]
3. State the mode of Abel’s score in a round of the game.[1]
4. Show that the variance of Abel’s score in a round of the game is \(\frac{60x+76}{3\left( 3x+1 \right)}-\frac{36{{\left( x+1 \right)}^{2}}}{{{\left( 3x+1 \right)}^{2}}}\).[3]
5. Given that \(x=3\) and that Abel plays \(100\) rounds of the game, find the probability that the average score is not less than \(2.5\).[3]