Solution: 2014 Fall Final - 17

Author: Michiel Smid

Question

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

and

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 = All possible outcomes
$ |S| = 10 $
A occurs when we choose an even number from the set ${2, 4, 6, 8, 10}$
$ |A| = 5 $
$Pr(A) = \frac{5}{10} = \frac{1}{2} $
B occurs when we choose a number from the set ${1, 2, 3, 4, 5}$
$ |B| = 5 $
$Pr(B) = \frac{5}{10} = \frac{1}{2} $
$A \cap B$ occurs when we choose an even number from the set ${2, 4}$
$ |A \cap B| = 2 $
$Pr(A \cap B) = \frac{2}{10} = \frac{1}{5} $

$ 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} $

Because the equation is false, these events are not independent.