a) The probability that the number of spots is even equals p_1 = 3/6 = 1/2, the probability that the number is either 1 or 3 equals p_2 = 2/6 = 1/3, and the probability that the number is 5 equals p_3 = 1/6. Hence, the expected value is 0*1/2 + 2*1/3 + 14 * 1/6 = (2+7)/3 = 3.

The square of the standard deviation is (0-3)^2*1/2 + (2-3)^2 * 1/3 + (14 - 3)^2 / 6 ) = 9/2 + 1/3 + 121/6

= (27 + 2 + 121)/6 = 150/6 = 25, hence, the standard deviation is 5.

b) If you play twice, there may be the following options:

probability winnings

both times you get an even number p = 1/4 0

both times you get either 1 or 3 p = 1/9 4

both times you get 5 p = 1/36 28

once you get an even number and

once you get either 1 or 3 p = 2/6 2

once you get an even number and

once you get 5 p = 2/12 14

once you get either 1 or 3

and once you get 5 p = 2/18 16

The expected value is 0*1/4 + 4*1/9 + 28*1/36 + 2*1/3 + 14*1/6 + 16*1/9 = 6 (twice the previous expected value) and the standard deviation is sqrt{50} = 7.07

c)

Apr 17th, 2015

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