Elementary Statistics work

timer Asked: Nov 8th, 2016

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QUESTION 1 (27 MARKS) A circular card is divided into 3 sectors scoring 1, 2 and 3 and having angles 1350, 900 and 1350 respectively. On a second circular card, sectors scoring 1, 2 and 3 have angles 1800, 900 and 900 respectively as in Diagram 1. Each card has a pointer pivoted at its centre. After being set in motion, the pointers come to rest independently in random position. 1 1 2 2 3 3 Diagram 1 a) The random variable 𝑋 and π‘Š are the scores obtain from circular card 1 and circular card 2 respectively. i. Construct the probability distribution of 𝑋 and π‘Š. ii. Find the probability that the score on each card is 1, (4 marks) iii. iv. (3 marks) Find the probability that the score on at least one of the cards is 3. (4 marks) Find the probability that the score on the second card is 2 given that the score on the first card is 3. (2 marks) b) The random variable π‘Œ is the larger of the two scores if they are different and their common value if they are the same. i. Construct a probability distribution for π‘Œ. (7 marks) ii. Show that 𝐸(π‘Œ) = 75 32 , (2 marks) iii. Find π‘‰π‘Žπ‘Ÿ(π‘Œ). (5 marks) QUESTION 2 (13 MARKS) Alfred and Betty play a game, each starting with cash amounting to RM100. Two dice are thrown. If the total score is 5 or more, then Alfred will pay RM π‘₯ to Betty where 0 < π‘₯ ≀ 8. If the total score is 4 or less, then Betty will pay RM (π‘₯ + 8)to Alfred. 1 a) Show that the expectation of Alfred cash after the first game is RM 3 (304 βˆ’ 2π‘₯). (8 marks) b) Find the value of π‘₯ for the game to be fair. (hint: the expectation of Alfred’s winnings equal to the expectation of Betty’s winning) (5 marks)

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