Solution: 2019 Fall Final - 18

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)

None of the above.

(b)

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

(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 6 numbers between 1 and 6: 6
$ |B| = 6 $
$ Pr(B) = \frac{6}{10} = \frac{3}{5} $
Let's determine $A \cap B$
There are 3 numbers that are both even and between 1 and 6, which are 2, 4, and 6: 2
$ |A \cap B| = 3 $
$ Pr(A \cap B) = \frac{3}{10} $

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Let’s check for independence

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

$ \frac{3}{10} = \frac{1}{2} \cdot \frac{3}{5} $

$ \frac{3}{10} = \frac{3}{10} $

Therefore, it is independent