Question

Cards marked with numbers 1, 3, 5,....., 101 are placed in a bag and mixed thoroughly. A card is drawn at random from the bag. Find the probability that the number on the drawn card is

(i) Less than 19,

(ii) a prime number less than 20.

Answer 1

Given number 1, 3, 5,,,,,,,101 form an AP with a = 1 and d = 2

Thus, total number of outcomes = 51

(i) Let E₁ be the event of getting a number less than 19.

Out of these numbers, number less than 19 are 1, 3 , 5,......, 17.

Given number 1, 3, 5,.....,17 form an AP with a = 1 and d = 2

Thus, the number of outcomes = 9

Therefore, P(getting a number less than 19) = P(E₁) = \(\frac{number\, of \,outcomes\,favorable\,to\,E_1}{number\,of \,all\,possible\,outcomes} \) = \(\frac{9}{51}\) = \(\frac{3}{17}\)

Thus, the probability that the number on the drawn card is less than 19 is \(\frac{3}{17}\).

(ii) Let E₂ be the event of getting a prime number less than 20.

Out of these numbers, prime number less than 20 are 3, 5, 7, 11, 13, 17 and 19.

Thus, the number of favorable outcomes = 7.

Therefore, P(getting a prime number less than 20) = P(E₂) = \(\frac{number\, of \,outcomes\,favorable\,to\,E_2}{number\,of \,all\,possible\,outcomes} \) = \(\frac{7}{51}\).

Thus, the probability that the number on the drawn card is a prime number less than 20 is \(\frac{7}{51}\).