Hundred identical cards are numbered from 1 to 100. The cards are well shuffled and then a card is drawn. Find the probability that the number on the card drawn is:

(i) a multiple of 5

(ii) a multiple of 6

(iii) between 40 and 60

(iv) greater than 85

(v) less than 48

We know that, there are 100 cards from which one card is drawn.

Number of possible outcomes = 100

(i) From numbers 1 to 100, there are 20 numbers which are multiple of 5 i.e. {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100}

∴ Number of favourable outcomes = 20.

P(selecting a card with a multiple of 5) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{20}{100} = \dfrac{1}{5}$.

Hence, the probability of selecting a card with a multiple of 5 = $\dfrac{1}{5}$.

(ii) From numbers 1 to 100, there are 16 numbers which are multiple of 6 i.e. {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96}

∴ Number of favourable outcomes = 16.

P(selecting a card with a multiple of 6) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{16}{100} = \dfrac{4}{25}$.

Hence, the probability of selecting a card with a multiple of 6 = $\dfrac{4}{25}$.

(iii) From numbers 1 to 100, there are 19 numbers which are between 40 and 60 i.e. {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59}

∴ Number of favourable outcomes = 19.

P(selecting a card between 40 and 60) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{19}{100}$.

Hence, the probability of selecting a card between 40 and 60 = $\dfrac{19}{100}$.

(iv) From numbers 1 to 100, there are 15 numbers which are greater than 85 i.e. {86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}

∴ Number of favourable outcomes = 15.

P(selecting a card with number greater than 85) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{15}{100} = \dfrac{3}{20}$.

Hence, the probability of selecting a card with number greater than 85 = $\dfrac{3}{20}$.

(v) From numbers 1 to 100, there are 47 numbers which are less than 48 i.e. {1, 2, ……….., 46, 47}

∴ Number of favourable outcomes = 47.

P(selecting a card with number less than 48) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{47}{100}$.

Hence, the probability of selecting a card with number less than 47 = $\dfrac{47}{100}$.