A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears

(i) a two-digit number

(ii) a perfect square number

(iii) a number divisible by 5

(iv) a prime number less than 30.

A disc is drawn at random from the box means that all the outcomes are equally likely.

Sample space = {1, 2, 3, ….., 90}, which has 90 equally likely outcomes.

(i) Let E₁ be the event of drawing a disc with two digit number.

E₁ = {10, 11, 12, 13, ……, 90}.

∴ The number of favourable outcomes to the event E₁ = 81.

$\therefore P(E$1) = \dfrac{\text{No. of favourable outcomes to } E1}{\text{Total no. of possible outcomes}} = \dfrac{81}{90} = \dfrac{9}{10}.

Hence, the probability of drawing a disc with two digit number is $\dfrac{9}{10}.$

(ii) Let E₂ be the event of drawing a disc with perfect square number.

E₂ = {1, 4, 9, 16, 25, 36, 49, 64, 81}.

∴ The number of favourable outcomes to the event E₂ = 9.

$\therefore P(E$2) = \dfrac{\text{No. of favourable outcomes to } E2}{\text{Total no. of possible outcomes}} = \dfrac{9}{90} = \dfrac{1}{10}.

Hence, the probability of drawing a disc with perfect square number is $\dfrac{1}{10}$.

(iii) Let E₃ be the event of drawing a disc with number divisible by 5.

E₃ = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90}.

∴ The number of favourable outcomes to the event E₃ = 18.

$\therefore P(E$3) = \dfrac{\text{No. of favourable outcomes to } E3}{\text{Total no. of possible outcomes}} = \dfrac{18}{90} = \dfrac{1}{5}.

Hence, the probability of drawing a disc with number that is divisible by 5 is $\dfrac{1}{5}$.

(iv) Let E₄ be the event of drawing a disc with prime number less than 30.

E₄ = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}.

∴ The number of favourable outcomes to the event E₄ = 10.

$\therefore P(E$4) = \dfrac{\text{No. of favourable outcomes to } E4}{\text{Total no. of possible outcomes}} = \dfrac{10}{90} = \dfrac{1}{9}.

Hence, the probability of drawing a disc with prime number less than 30 is $\dfrac{1}{9}$.