Question The value of Z for each X, Y here is obtained as:

 X Y Z 1 1 2 1 2 3 1 3 4 1 4 5 2 1 3 2 2 4 2 3 5 2 4 6 3 1 4 3 2 5 3 3 6 3 4 7 4 1 5 4 2 6 4 3 7 4 4 8

Using this the PDF for Z here is obtained as:

 z P(z) 2 0.0625 3 0.125 4 0.1875 5 0.25 6 0.1875 7 0.125 8 0.0625

a) The expected value of Z here is computed as:  Therefore 5 is the expected value here.

b) Using bayes theorem, probability here is computed as:
P(Z = 7 | Z > 5) = P(Z = 7) / P(Z = 6, 7 or 8)

= 0.125 / (0.1875 + 0.125 + 0.0625)

= 0.35

therefore 0.35 is the required probability here.

c) The probability that we will have to play 4 games before we roll Z = 7 is computed here as:

= [1 - P(Z = 7) ]4 = (1 - 0.125)4 = 0.5862

Therefore 0.5862 is the required probability here.

d) The probability that in 10 games played, we get Z = 4 five times is computed here using binomial probability function as: Therefore 0.0207 is the required probability here.

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