Consider the following game. A person rolls 14 fair six-sided die. Each roll of

Question

Consider the following game. A person rolls 14 fair six-sided die. Each roll of six is worth 5 points, but every other number costs 1 point. That is, the reward R is 5 times the number of sixes minus the number of non-sixes. (Note that R can be negative). Let Y be the random variable on this game which just counts the number of sixes rolled. (A) Find n and p such that Y is equivalent to Bin(n, p). (B) Express R as a formula in Y. (C) What is the smallest value that R can return? What is the greatest? (D) What is the probability that Y = 4? (E) What is the value of R when Y = 4? (F) What is the expected value of R?

Explanation / Answer

(a) Here n = 14 and p = 1/6 as there are 6 choices and we will get 6 as one choice.

so Y ~ Bin (14,1/6)

(B) R = 5Y - 1 * (14 -Y) = 5Y - 14 + Y = 6Y - 14

(C) Smallest value of R will be when Y = 0 and Rmin = -14

Largest value of R will be when Y = 14 so Rmax = 70

(D) Probability that Y = 4

Pr (Y =4) = Nin (4; 14; 1/6) = 14C 4 (1/6)4 (5/6)10 = 0.125

(E) value of R when Y = 4

R = 6Y - 14 = 6 * 4 - 14 = 10

(F) Expected Value of R = ?

Expected value of Y = E(Y) = 14 * (1/6) = 2.33

Expected Value of R = 6 * E(Y) - 14 = 6 * 2.33 - 14 = 0

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