Solution: 2022 Winter Final - 21

Author: Michiel Smid

Question

Let $D_1$ and $D_2$ be the result of tossing a $6$-sided die twice. Define the random variables \[ X=\max\{D_1,D_2\} \] and \[ Y=D_1+D_2 \enspace . \] Which of the following is true?

(a)

$X$ and $Y$ are not independent

(b)

$X$ and $Y$ are independent

(c)

None of the above

Solution

Forget making tables for this. I don’t wanna

Now, let’s see if there are any $\Pr(X=i)$ and $\Pr(Y=j)$ that don’t have a product of 0, yet $\Pr(X_i \cap Y_j) = 0$

$ \Pr(X=1) $ is asking for rolls that have max values of 1: $ (1,1) $

$\Pr(X=1) = \frac{1}{36} $

$ \Pr(Y=12) $ is asking for rolls with a sum of 12: $ (6,6) $

$ \Pr(Y=12) = \frac{1}{36} $

$ \Pr(X=1 \cap Y=12) $ is asking for rolls that have max values of 1 yet produce a sum of 12. This is impossible. 1 and 1 don’t make 12, so the probability is 0

$ \Pr(X=1 \cap Y=12) = 0$

As you can see, it isn’t independence day