Cards marked with numbers 1, 2, 3, 4, ……, 20 are well-shuffled and a card is drawn at random. What is the probability that the number on the card is :

(i) a prime number

(ii) divisible by 3

(iii) a perfect square?

Cards are marked 1, 2, 3, 4, ……, 20 and a card is drawn at random.

Sample space = {1, 2, 3, 4, ……, 20}, which has 20 equally likely outcomes.

(i) Let E₁ be the event of choosing a prime number.

E₁ = {2, 3, 5, 7, 11, 13, 17, 19}.

∴ The number of favourable outcomes to the event E₁ = 8.

$\therefore P(E$1) = \dfrac{\text{No. of favourable outcomes to } E1}{\text{Total no. of possible outcomes}} = \dfrac{8}{20} = \dfrac{2}{5}.

Hence, the probability of choosing a prime number is $\dfrac{2}{5}$.

(ii) Let E₂ be the event of choosing a number that is divisible by 3.

E₂ = {3, 6, 9, 12, 15, 18}.

∴ The number of favourable outcomes to the event E₂ = 6.

$\therefore P(E$2) = \dfrac{\text{No. of favourable outcomes to } E2}{\text{Total no. of possible outcomes}} = \dfrac{6}{20} = \dfrac{3}{10}.

Hence, the probability of choosing a number divisible by 3 is $\dfrac{3}{10}$.

(iii) Let E₃ be the event of choosing a perfect square.

E₃ = {1, 4, 9, 16}.

∴ The number of favourable outcomes to the event E₃ = 4.

$\therefore P(E$3) = \dfrac{\text{No. of favourable outcomes to } E3}{\text{Total no. of possible outcomes}} = \dfrac{4}{20} = \dfrac{1}{5}.

Hence, the probability of choosing a perfect square is $\dfrac{1}{5}$.