A box contains cards bearing numbers 6 to 70. If one card is frawn at random from the box, find the probability that it bears (i) a one-digit number, (ii) a number divisible by 5, (iii) an odd number less than 30, (iv) a composite number between 50 and 70.

Answer

`(i) 4/65 (ii) 1/5 (iii) 12/65 (iv) 3/25`

Explanation

Given numbers 6,7,8,…,70 form an AP with a = 6 and d = 1.
Let` T_(n) = 70." Then, " 6+(n-1) xx 1 = 64 rArr n = 65`.
` :. ` total numbers of cards = 65.
(i) Out of the given numbers, the one-digit numbers are 6,7,8,9.
Number of one-digit numbers = 4.
` :. ` P(getting a one-digit number) =` 4/65`.
(ii) Out of the given numbers, those divisible by 5 are` 10,15,20,25,...,70`.
Let ` T_(n) = 70.` Then,
` 10 + (n-1) xx 5 = 70 rArr (n-1) xx 5 = 60 rArr n -1 = 12 rArr n = 13`.
` :. ` p(getting a number divisible by 5) ` = 13/65 = 1/5`.
(iii) Out of the given numbers, odd numbers less than 30 are 7,9,11,13,...,29.
Let ` T_(n) = 29 `. Then,
` 7 +(m-1) xx 2 = 29 rArr (m-1) xx 2 = 22 rArr m -1 = 11 rArr m = 12`.
` :.` P(getting an odd number less than 30) = ` 12/65`.
(iv) Number of numbers between 50 and 70 = numbers from 51 to 69.
Their number = ` (69-51) + 1 = 19`.
Prime number between 50 and 70 = 53, 59, 61, 67.
Number of prime numbers = 4.
Number of composite numbers = 19-4 = 15.
` :.` P(getting a composite numbers) ` = 15/65 = 3/13`.