A box contains 15 balls bearing numbers 1, 2, 3, …, 14, 15 respectively. A ball is drawn at random from the box. Find the probability that the number on the ball is:

(i) an even number

(ii) a number divisible by 5

(iii) the number 6

(iv) a number lying between 8 and 12

(v) a number greater than 9

(vi) a number less than 6

Given,

Balls in the box are numbered

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

Total number of outcomes = 15

(i) Let A be the event of getting an even number, then

A = {2, 4, 6, 8, 10, 12, 14}

∴ The number of favourable outcomes to the event A = 7

∴ P(A) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{7}{15}$

Hence, the probability of getting an even number is $\dfrac{7}{15}$.

(ii) Let B be the event of getting a number divisible by 5, then

B = {5, 10, 15}

∴ The number of favourable outcomes to the event B = 3

∴ P(B) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{3}{15} = \dfrac{1}{5}$

Hence, the probability of getting a number divisible by 5 is $\dfrac{1}{5}$.

(iii) Let C be the event of getting the number 6, then

C = {6}

∴ The number of favourable outcomes to the event C = 1

∴ P(C) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{1}{15}$

Hence, the probability of getting the number 6 is $\dfrac{1}{15}$.

(iv) Let D be the event of getting a number between 8 and 12, then

D = {9, 10, 11}

∴ The number of favourable outcomes to the event D = 3

∴ P(D) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{3}{15} = \dfrac{1}{5}$

Hence, the probability of getting a number lying between 8 and 12 is $\dfrac{1}{5}$.

(v) Let E be the event of getting a number greater than 9, then

E = {10, 11, 12, 13, 14, 15}

∴ The number of favourable outcomes to the event E = 6

∴ P(E) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{6}{15} = \dfrac{2}{5}$

Hence, the probability of getting a number greater than 9 is $\dfrac{2}{5}$.

(vi) Let F be the event of getting a number less than 6, then

F = {1, 2, 3, 4, 5}

∴ The number of favourable outcomes to the event F = 5

∴ P(F) = $\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{5}{15} = \dfrac{1}{3}$

Hence, the probability of getting a number less than 6 is $\dfrac{1}{3}$.