Question

A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that, number on the ticket is a prime number or greater than 40

Answer 1

Out of the 75 tickets, one ticket can be drawn in ⁷⁵C₁ = 75 ways.

∴ n(S) = 75

Let event A: The number on the ticket is 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}

∴ n(A) = 21

∴ P(A) = `("n"("A"))/("n"("S")) = 21/75`

Let event B: The number is greater than 40.

∴ B = {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75}

∴ n(B) = 35

∴ P(B) = `("n"("B"))/("n"("S")) = 35/75`

Now, A ∩ B = {41, 43, 47, 53, 59, 61, 67, 71, 73}

∴ n(A ∩ B) = 9

∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 9/75`

∴ Required probability = P(A ∪ B) 

= P(A) + P(B) – P(A ∩ B)

= `21/75 + 35/75 - 9/75`

= `47/75`.