1973 TYS

A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each correct answer a candidate gets $2$ marks, but loses $1$ mark for each incorrect answer. A particular candidate answers all question purely by guesswork. Draw up a table to show all the possible total marks from $-5$ to $10$ and the probability associated with each mark for this candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.

1. In a game, a player throws an ordinary unbiased die. If it is not six, he scores the number on the upper face of the die; if he throws a six, he has a second throw and his score is then the total obtained from his two throw. (i.e. The player has at most two throws.)
   Find the probability distribution function of his score.

   [2]

   Calculate
   1. The expected value and standard deviation of the player’s score,

      [2]

   2. The expected value of the number of throws.

      [2]

2. The player pays $\$2$ to play a game. If he scores an even number, he will receive $\$3.50$, else he gets back $\$1$.
   1. Determine if this is a fair game.

      [2]

   2. Find the probability that the player’s score is an even number in $20$ out of total of $50$ games played.

      [2]