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Question

A player throws a fair cubical dice whose faces are numbered 1 to 6 inclusive. If the player obtains a 6 he throws the dice a second time, and in this case his score is the sum of 6 and the second number: otherwise his score is the number obtained. The player has no more than two throws.
Let X be the random variable denoting the player's score. Write down the probability distribution of X and determine its mean and standard deviation.

Answer 1

Let f(x) be the probability mass function of X. Then:

f(x) = {1/6 if 1≤x≤5 (x∈N)
1/36 if 7≤x≤12 (x∈N)}

This is because the probability of scoring 1-5 is simply the probability of rolling that particular number (1/6). The probability of scoring 7-12 is the probability of rolling a 6 on the first throw, and n-6 on the second (1/6 * 1/6). It is impossible to score exactly six, since the second throw will add at least one to the score.

Now, let us find the first and second moments:

∑x f(x) = (1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 7/36 + 8/36 + 9/36 + 10/36 + 11/36 + 12/36) = 147/36 = 49/12

∑x² f(x) = (1/6 + 4/6 + 9/6 + 16/6 + 25/6 + 49/36 + 64/36 + 81/36 + 100/36 + 121/36 + 144/36) = 889/36

The mean of this distribution is the first moment, which is 49/12.

The standard deviation is the square root of the variance, which is in turn the difference between the second moment and the square of the first moment. So:

σ² = 889/36 - (49/12)² = 385/48
σ = √(385/48)

Answer 2

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Answer 3

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Answer 4

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