1. 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?

2. Originally Posted by Sally_Math
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: $\displaystyle x_1, x_2$. And we know that $\displaystyle x_1,x_2 \in [1,2,3,4,5,6]$.

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

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

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

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