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Math Help - probability question

  1. #1
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    Exclamation probability question

    I am totally confused on how to solve this problem, and I don't have any data.

    A fair die is rolled twice with the two rolls being independent of each other. Let M be the maximum of the two rolls and D be the value of the first roll minus the value of the second roll. Are M and D independent?
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  2. #2
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    Quote Originally Posted by Sally_Math View Post
    I am totally confused on how to solve this problem, and I don't have any data.

    A fair die is rolled twice with the two rolls being independent of each other. Let M be the maximum of the two rolls and D be the value of the first roll minus the value of the second roll. Are M and D independent?
    Denote the values of the two rolls, respectively, to be: x_1, x_2. And we know that x_1,x_2 \in [1,2,3,4,5,6].

    An intuitive method for seeing the answer is to consider the extreme cases.

    Say that M=1. This implies that the max(x_1,x_2)=1, which means that x_1 \leq 1 and x_2 \leq 1. So then it must be the case that x_1 = x_2 = 1. Then you know that D= x_2 - x_1 = 1-1 = 0. As a result, knowing that M=1 gives you information about D, specifically that D must be zero.

    Now say that M=6. That implies that max(x_1,x_2)=6 \rightarrow x_1, x_2 \leq 6. That means that you could have D=5 (in the case where x_2=6,x_1=1), or you could have D=0 (in the case where x_2=6,x_1=6) or anything in between. Less information is known about the value of D now, when M takes a larger value.

    When M=1, you are sure that D=0 but when M=6, D could be anything from 0 to 5. So the smaller that M is, the more certain you are about the values that D can take. By definition, then, because the value of D depends on the value of M, they cannot be independent.
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