There are 25 discs numbered 1 to 25. They are put in a closed box and shaken throughly. A disc is drawn at random from the box. Find the probability that the number on the disc is:

(i) an odd number

(ii) divisible by 2 and 3 both

(iii) a number less than 16.

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

E₁ = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}.

∴ The number of favourable outcomes to the event E₁ = 13.

$\therefore P(E$1) = \dfrac{\text{No. of favourable outcomes to } E1}{\text{Total no. of possible outcomes}} = \dfrac{13}{25}.

Hence, the probability of choosing an odd number disc is $\dfrac{13}{25}$.

(ii) Let E₂ be the event of choosing a disc with number that is divisible by both 2 and 3.

E₂ = {6, 12, 18, 24}.

∴ The number of favourable outcomes to the event E₂ = 4.

$\therefore P(E$2) = \dfrac{\text{No. of favourable outcomes to } E2}{\text{Total no. of possible outcomes}} = \dfrac{4}{25}.

Hence, the probability of choosing a disc with number that is divisible by both 2 and 3 is $\dfrac{4}{25}$.

(iii) Let E₃ be the event of choosing a disc with number less than 16.

E₃ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

∴ The number of favourable outcomes to the event E₃ = 15.

$\therefore P(E$3) = \dfrac{\text{No. of favourable outcomes to } E3}{\text{Total no. of possible outcomes}} = \dfrac{15}{25} = \dfrac{3}{5}.

Hence, the probability of choosing a disc with number less than 16 is $\dfrac{3}{5}$.