Solution: 2019 Winter Final - 16

Author: Michiel Smid

Question

You roll a fair red die once and you roll a fair blue die once. These two rolls are independent. Consider the events

Which of the following is correct?

(a)

The events $A$ and $B$ are independent.

(b)

None of the above.

(c)

All of the above.

(d)

The events $A$ and $B$ are not independent.

Let’s determine A

As can be seen, A occurs 4 out of the 36 possible outcomes.

$ \Pr(A) = \frac{4}{36} $
Let’s determine B

The red die is even half the time

$ \Pr(B) = \frac{1}{2} $
Let’s determine $ \Pr(A \cap B) $

$A \cap B = {(2,3), (4,1)}$

$|A \cap B| = 2$

$ \Pr(A \cap B) = \frac{2}{36} $
Profit

Check if A and B are independent.

$ \Pr(A \cap B) = \Pr(A) \cdot \Pr(B) $

$ \frac{2}{36} = \frac{4}{36} \cdot \frac{1}{2} $

$ \frac{2}{36} = \frac{2}{18} \cdot \frac{1}{2} $

$ \frac{2}{36} = \frac{2}{36} $