N1973 P2 Q11 [Modified]

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]