Solution: 2017 Winter Final - 21

Author: Michiel Smid

Question

You are given a fair red die and a fair blue die. You roll each die once, independently of each other. Let $(i,j)$ be the outcome, where $i$ is the result of the red die and $j$ is the result of the blue die. Define the random variables $$ X = |i - j| $$ and $$ Y = \max(i, j). $$ Which of the following is true?

(a)

The random variables $X$ and $Y$ are not independent.

(b)

The random variables $X$ and $Y$ are independent.

(c)

None of the above.

Solution

We don’t need to calculate all possibilities. Since this is independence, we’re looking for $ Pr(X=x \cap Y=y) = Pr(X=x) \cdot Pr(Y=y) $

Let's try $X=5$
The only rolls that satisfy this are $ (1,6), (6,1) $
$ Pr(X=5) = \frac{2}{36} = \frac{1}{18} $
Let's try $Y=1$
The only rolls that satisfy this are $ (1,1) $
$ Pr(Y=1) = \frac{1}{36} $
$ Pr(X=5 \cap Y=1) $
If the only rolls that satisfy condition 1 are $ (1,6), (6,1) $ and the only rolls that satisfy condition 2 are $ (1,1) $, then there are no rolls that satisfy both conditions
$ Pr(X=5 \cap Y=1) = 0 $

Now, check time

$ Pr(X=5 \cap Y=1) = Pr(X=5) \cdot Pr(Y=1) $

$ 0 = \frac{1}{18} \cdot \frac{1}{36} $

$ 0 = \frac{1}{648} $

Since the equation is false, there are dependent