Solution: 2019 Fall Final - 17

Author: Michiel Smid

Question

We choose an element $x$ uniformly at random from the set $\{1,2,3,...,10\}$. Define the events

Which of the following is true?

(a)

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

(b)

None of the above.

(c)

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

Let's determine $S$
$ |S| = 10 $
Let's determine $A$
There are 5 even numbers: 5
$ |A| = 5 $
$ Pr(A) = \frac{5}{10} = \frac{1}{2} $
Let's determine $B$
There are 5 numbers between 1 and 5: 5
$ |B| = 5 $
$ Pr(B) = \frac{5}{10} = \frac{1}{2} $
Let's determine $A \cap B$
There are 2 numbers that are both even and between 1 and 5, which are 2 and 4: 2
$ |A \cap B| = 2 $
$ Pr(A \cap B) = \frac{2}{10} = \frac{1}{5} $

Avatar: The Last Airbender is pretty high tier

Let’s check for independence

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

$ \frac{1}{5} = \frac{1}{2} \cdot \frac{1}{2} $

$ \frac{1}{5} = \frac{1}{4} $

The equation is not false; therefore, they’re dependent