Find the probability that a number selected at random from the numbers 1, 2, 3, ……, 35 is a

(i) prime number

(ii) multiple of 7

(iii) multiple of 3 or 5

A number is selected at random it means that all outcomes are equally likely.

Sample space = {1, 2, 3, ……, 35}, which has 35 equally likely outcomes.

(i) Let A be the event 'the number is prime', then

A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}.

∴ The number of favourable outcomes to the event A = 11.

∴ P(prime number) = $\dfrac{11}{35}.$

Hence, the probability that the number selected is a prime number is $\dfrac{11}{35}.$

(ii) Let B be the event 'the number is multiple of 7', then

B = {7, 14, 21, 28, 35}.

∴ The number of favourable outcomes to the event B = 5.

∴ P(multiple of 7) = $\dfrac{5}{35} = \dfrac{1}{7}.$

Hence, the probability that the number selected is a multiple of 7 is $\dfrac{1}{7}.$

(iii) Let C be the event 'the number is multiple of 3 or 5', then

C = {3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30, 33, 35}.

∴ The number of favourable outcomes to the event C = 16.

∴ P(multiple of 3 or 5) = $\dfrac{16}{35}.$

Hence, the probability that the number selected is a multiple of 3 or 5 is $\dfrac{16}{35}.$