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 perfect square or divisible by 4

#### Solution

One ticket can be drawn out of 75 tickets in ^{75}C_{1} = 75 ways.

∴ n(S) = 75

Let A ≡ the event that number on the ticket is a perfect square

∴ A = {1, 4, 9, 16, 25, 36, 49, 64}

∴ n(A) = 8

∴ P(A) = `("n"("A"))/("n"("S")) = 8/75`

Let B ≡ the event that number on the ticket is divisible by 4

∴B = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72}

∴ n(B) = 18

∴ P(B) = `("n"("B"))/("n"("S")) = 18/75`

Now, A ∩ B = {4, 16, 36, 64}

∴ n(A ∩ B) = 4

∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 4/75`

∴ the required probability = P(A ∪ B)

= P(A) + P(B) – P(A ∩ B)

= `8/75 + 18/75 - 4/75`

= `22/75`