Question

Cards numbered from 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn cards is (i) an odd number (ii) a perfect square number (iii) divisible by 5. (iv) a prime number less than 20.

Answer 1

All possible outcomes are 11,12,13,…60.
Number of all the event that the number on the drawn card is an old number.
Then, the favourable outcomes are 11,13,15,…,59.
Clearly, these numbers form an AP with a = 11 and d = 2.
Let the number of these be n. Then,
`T_(n) = 59 rArr 11+(n-1) xx 2 = 59 rArr (n-1) xx 2 = 48`
` rArr (n-1) = 24 rArr n = 25`.
So, the number of favourable outcomes = 25.
`:. ` P(getting an old number) = `P(E_(1)) = 25/50 = 1/2`.
(ii) Let `E_(2)` be the event that the number on the drawn card is a perfect square number.
Then, the favourable outcomes are 16, 25, 36, 49.
So, the number of favourable outcomes = 4.
`:. ` P(getting a perfect square number) = `P(E_(2)) = 4/50 = 2/25`.
(iii) Let ` E_(3)` be the event that the number on the drawn card is divisible by 5.
Then, the favourable outcomes are 15, 20, 25,....,60.
Clearly, these numbers form an AP with a = 15 and d = 5.
Let the number of these terms be m. Then,
`T_(m) = 60 rArr 15+(m-1) xx 5 = 60 rArr (m-1) xx 5 = 45`
` rArr m -1 = 9 rArr m = 10`.
So, the number of favourable outcomes = 10.
`:. ` P(getting a number divisible by 5) =`P(E_(3)) = 10/50 = 1/5`.
(iv) Let `E_(4)` be the event that the number on the drawn card is a prime number less than 20.
Then, the favourable outcomes are 11,13,17,19.
So, the number of favourable outcomes = 4.
`:. ` P(getting a prime number less than 20) = `P(E_(4)) = 4/50 = 2/25`.