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 less than 19.

Answer 1

​Given number 1, 3, 5, ..., 101 form an AP with a = 1 and d = 2.
Let T_n = 101. Then,
1 + (n − 1)2 = 101
⇒ 1 + 2n  − 2 = 101
⇒ 2n = 102
⇒ n = 51

Thus, total number of outcomes = 51.

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

Out of these numbers, numbers less than 19 are 1, 3, 5, ... , 17.
Given number 1, 3, 5, .... , 17 form an AP with a = 1 and d = 2.
Let T_n = 17. Then,
1 + (n − 1)2 = 17
⇒ 1 + 2n  − 2 = 17
⇒ 2n = 18
⇒ n = 9

Thus, number of favourable outcomes = 9.

∴ P(getting a number less than 19) = P(E₁) = `("Number of outcomes favourable to" E_2)/"Number of all possible outcomes"`

Thus, the probability that the number on the drawn card is less than 19 is `3/17`.