A test consists of five multiple choice questions to each of which three answers are given, only one

Question

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. 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 The expected value and standard deviation of the player’s score, [2] The expected value of the number of throws. [2] The player pays $\$2$ to play a game. If he scores an even number, he will receive $\$3.50$, else he gets back $\$1$. Determine if this is a fair game. [2] Find the probability that the player’s score is an even number in $20$ out of total of $50$ games played. [2]

Accepted Answer

(a)(i) $E(X) = \frac{49}{12}$, $\text{Var}(X) = \frac{385}{48}$ (a)(ii) $E(T) = \frac{7}{6}$ (b)(i) Expected profit $\frac{1}{24}$, not fair (b)(ii) $0.111$

N1973 P2 Q11 [Modified]

N1973 P2 Q11 [Modified]

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