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Solution: 2013 Fall Midterm - 16

Author: Pat Morin

Question

If you choose an element $ x $ uniformly at random from the set $ {1, 2, ..., 100} $, what is the probability that $ x $ is divisible by 4 or 5?

(a)

$2/5$

(b)

$9/100$

(c)

$1/5$

(d)

$45/100$

Solution

Let’s break this down into two cases.

Determine A

A = divisible by 4

$A = { 4,8,12,…,96,100 }$

$ |A| = 25 $
Determine B

B = divisible by 5

$B = { 5,10,15,…,95,100 }$

$ |B| = 20 $
Determine the Intersection

$ |A \cap B| = 5 $
Profit

$ |A \cup B| = |A| + |B| - |A \cap B| $

$ |A \cup B| = 25 + 20 - 5 $

$ |A \cup B| = 40 $

Pr($A \cup B$) = $ \frac{40}{100} $

Pr($A \cup B$) = $ \frac{2}{5} $