Solution: 2017 Fall Final - 16

Author: Michiel Smid

Question

You choose a uniformly random element, say $x$, from the set $\{1,2,3\}$, and you choose a uniformly random element, say $y$, from the same set $\{1,2,3\}$. ($x$ and $y$ are chosen independently of each other.) Define the events

A = "$x$ is odd",
B = "$x + y$ is odd".

Which of the following is true?

(a)

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

(b)

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

(c)

None of the above.

Solution

I kind of brute forced a tree

Now, let’s determine some variables

Let's determine A
$ { (1,1), (1,2), (1,3), (3,1), (3,2), (3,3) } $
$ |A| = 6 $
$ Pr(A) = \frac{6}{9} $
Let's determine B
$ { (1,2), (2,1), (2,3), (3,2) } $
$ |B| = 4 $
$ Pr(B) = \frac{4}{9} $
Let's determine $ A \cap B $
$ { (1,2), (3,2) } $
$ |A \cap B| = 2 $
$ Pr(A \cap B) = \frac{2}{9} $

Now, let’s check if A and B are independent

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

$ \frac{2}{9} = \frac{6}{9} \cdot \frac{4}{9} $

$ \frac{2}{9} = \frac{24}{81} $

$ \frac{2}{9} = \frac{8}{27} $

Since the equation is false, A and B are not independent.