A box contains 15 cards numbered 1, 2, 3, …., 15 which are mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the card is :

(i) odd

(ii) prime

(iii) divisible by 3

(iv) divisible by 3 and 2 both

(v) divisible by 3 or 2

(vi) a perfect square number.

(i) Let E₁ be the event of choosing an odd number card.

E₁ = {1, 3, 5, 7, 9, 11, 13, 15}.

∴ 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}{15}.

Hence, the probability of choosing an odd number card is $\dfrac{8}{15}$.

(ii) Let E₂ be the event of choosing a prime number card.

E₂ = {2, 3, 5, 7, 11, 13}.

∴ 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}{15} = \dfrac{2}{5}.

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

(iii) Let E₃ be the event of choosing card with number that is divisible by 3.

E₃ = {3, 6, 9, 12, 15}.

∴ The number of favourable outcomes to the event E₃ = 5.

$\therefore P(E$3) = \dfrac{\text{No. of favourable outcomes to } E3}{\text{Total no. of possible outcomes}} = \dfrac{5}{15} = \dfrac{1}{3}.

Hence, the probability of choosing a card with number that is divisible by 3 is $\dfrac{1}{3}$.

(iv) Let E₄ be the event of choosing card with number that is divisible by 3 and 2.

E₄ = {6, 12}.

∴ The number of favourable outcomes to the event E₄ = 2.

$\therefore P(E$4) = \dfrac{\text{No. of favourable outcomes to } E4}{\text{Total no. of possible outcomes}} = \dfrac{2}{15}.

Hence, the probability of choosing a card with number that is divisible by 3 and 2 is $\dfrac{2}{15}$.

(v) Let E₅ be the event of choosing card with number that is divisible by 3 or 2.

E₅ = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}.

∴ The number of favourable outcomes to the event E₅ = 10.

$\therefore P(E$5) = \dfrac{\text{No. of favourable outcomes to } E5}{\text{Total no. of possible outcomes}} = \dfrac{10}{15} = \dfrac{2}{3}.

Hence, the probability of choosing a card with number that is divisible by 3 or 2 is $\dfrac{2}{3}$.

(vi) Let E₆ be the event of choosing card with perfect square number.

E₆ = {1, 4, 9}.

∴ The number of favourable outcomes to the event E₆ = 3.

$\therefore P(E$6) = \dfrac{\text{No. of favourable outcomes to } E6}{\text{Total no. of possible outcomes}} = \dfrac{3}{15} = \dfrac{1}{5}.

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