A bag contains 100 identical marble stones which are numbered from 1 to 100. If one stone is drawn at random from the bag, find the probability that it bears :

(i) a perfect square number.

(ii) a number divisible by 4.

(iii) a number divisible by 5.

(iv) a number divisible by 4 or 5.

(v) a number divisible by 4 and 5.

There are 100 identical marble stones.

∴ No. of possible outcomes = 100.

(i) Stones containing a perfect square number are numbered :

1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

No. of favourable outcomes = 10.

P(drawing a stone with perfect square number)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{10}{100} = \dfrac{1}{10}$.

Hence, the probability of drawing a stone bearing a perfect square number = $\dfrac{1}{10}$.

(ii) Stones containing a number which is divisible by 4 are numbered :

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

No. of favourable outcomes = 25.

P(drawing a stone with a number divisible by 4)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{25}{100} = \dfrac{1}{4}$.

Hence, the probability of drawing a stone with a number divisible by 4 = $\dfrac{1}{4}$.

(iii) Stones containing a number which is divisible by 5 are numbered :

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100.

No. of favourable outcomes = 20.

P(drawing a stone with a number divisible by 5)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{20}{100} = \dfrac{1}{5}$.

Hence, the probability of drawing a stone with a number divisible by 5 = $\dfrac{1}{5}$.

(iv) Stones containing a number which is divisible by 4 or 5 are numbered :

4, 8, 12, 16, 24, 28, 32, 36, 44, 48, 52, 56, 64, 68, 72, 76, 84, 88, 92, 96, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100.

No. of favourable outcomes = 40.

P(drawing a stone with a number divisible by 4 or 5)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{40}{100} = \dfrac{2}{5}$.

Hence, the probability of drawing a stone with a number divisible by 4 or 5 = $\dfrac{2}{5}$.

(v) Stones containing a number which is divisible by 4 and 5 are numbered :

20, 40, 60, 80, 100.

No. of favourable outcomes = 5.

P(drawing a stone with a number divisible by 4 and 5)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{5}{100} = \dfrac{1}{20}$.

Hence, the probability of drawing a stone with a number divisible by 4 and 5 = $\dfrac{1}{20}$.