Question

An integer is chosen at random from the first 100 positive integers. What is the probability that the integer chosen is a prime or multiple of 8?

Answer 1

Let S be the sample space

S = {1, 2, 3, …………., 100}

Let A be the event of choosing a prime number

A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}

n(A) = 25

Let B be the event of choosing an integer a multiple of 8.

B = {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96}

n(B) = 12

P(Choosing a prime number) = P(A)

= `("n"("A"))/("n"("S"))`

= `25/100`

= `1/4`

P(Choosing an integer a multiple of 8) = P(B)

= `("n"("B"))/("n"("S"))`

= `12/100`

P(Choosing an integer a prime or multiple of 8)

= P(A or B)

= P(A ∪ B)

P(A) + P(B)

(since A and B are mutually exclusive that is A ∩ B = Φ)

= `("n"("A"))/("n"("S"))  + ("n"("B"))/("n"("S"))`

= `25/100 + 12/100`

=  `(25 + 12)/100`

= `37/100`