A fair four-sided dice, numbered 1, 2, 3, and 4 is rolled twice. If the score on

Question

A fair four-sided dice, numbered 1, 2, 3, and 4 is rolled twice. If the score on the 2nd roll is strictly greater than the score on the first roll, then the player wins the difference in dollars. If the score on the second roll is strictly less than the score on the 1st roll, then the player loses the difference in dollars. If the scores are equal, the player neither wins nor loses. If we let X denote the (possibly negative) winnings of the player, what is the probability mass function of X? (Note that X can take any of the values from {-3, -2, -1, 0, 1, 2, 3}).

Explanation / Answer

X denotes the winning. {a,b} denotes {1st roll score,2nd roll score}

(X=-3) = {4,1}

(X=-2) = {4,2}, {3,1}

(X=-1) = {4,3}, {3,2}, {2,1}

(X=0) = {4,4}, {3,3}, {2,2}, {1,1}

(X=1) = {3,4}, {2,3}, {1,2}

(X=2) = {2,4}, {3,1}

(X=3) = {1,4}

Probability Mass Function(X=x) .

P(X=-3) = (0.25 * 0.25) = 0.0625

P(X=-2) = (0.25 * 0.25) * 2 = 0.125

P(X=-1) = (0.25 * 0.25) * 3 = 0.1875

P(X=0) = (0.25 * 0.25) * 4 = 0.25

P(X=1) = (0.25 * 0.25) * 3 = 0.1875

P(X=2) = (0.25 * 0.25) * 2 = 0.125

P(X=3) = (0.25 * 0.25) * 1 = 0.0625

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