Question

A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that, 

(i) number on the ticket is a perfect square or divisible by 4. 

(ii) 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 

(i) Let event A: The number on the ticket is a perfect square. 

∴ A = {1, 4, 9, 16, 25, 36, 49, 64} 

∴ n(A) = 8

∴ P(A) = \(\frac {n(A)}{n(S)} = \frac 8{75}\)

Let event B: The 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) = \(\frac {n(B)}{n(S)} = \frac {18}{75}\)

∴ Required probability 

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

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

= 22/75

(ii) 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) = \(\frac {n(A)}{n(S)} = \frac {21}{75}\)

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) = \(\frac {n(B)}{n(S)} = \frac {35}{75}\)

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

∴ n(A ∩ B) = 9