From a well-shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn is :

(i) a face card

(ii) not a face card

(iii) a queen of black colour

(iv) a card with number 5 or 6

(v) a card with number less than 8

(vi) a card with number between 2 and 9

There are 52 cards in a deck.

We have, the total number of possible outcomes = 52

(i) No. of face cards in a deck of 52 cards = 12 (4 kings, 4 queens and 4 jacks)

∴ No. of favourable outcomes = 12

P(drawing a face card) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{12}{52} = \dfrac{3}{13}$.

Hence, probability of drawing a face card = $\dfrac{3}{13}$.

(ii) As, probability of drawing a face card and a non-face card are complimentary event.

∴ Probability of drawing a face card + Probability of drawing a non-face card = 1

⇒ Probability of not drawing a face card = 1 - Probability of drawing a face card

⇒ Probability of not drawing a face card = $1 - \dfrac{3}{13} = \dfrac{13 - 3}{13} = \dfrac{10}{13}$.

Hence, probability of not drawing a face card = $\dfrac{10}{13}$.

(iii) There are 2 queens of black colour (1 of each club and spade).

∴ No. of favourable outcomes = 2

P(drawing a queen of black colour)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{2}{52} = \dfrac{1}{26}$.

Hence, probability of drawing a queen of black colour = $\dfrac{1}{26}$.

(iv) There are 4 cards (1 of each suit) of each 5 and 6 number.

∴ No. of favourable outcomes = 8

P(drawing a card with number 5 or 6)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{8}{52} = \dfrac{2}{13}$.

Hence, probability of drawing a card with number 5 or 6 = $\dfrac{2}{13}$.

(v) There are {2, 3, 4, 5, 6, 7} numbered cards of each heart, diamond, club and spades.

∴ No. of favourable outcomes = 6 × 4 = 24.

P(getting a card with number less than 8)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{24}{52} = \dfrac{6}{13}$.

Hence, the probability of drawing a card with number less than 8 = $\dfrac{6}{13}$.

(vi) There are {3, 4, 5, 6, 7, 8} numbered cards of each heart, diamond, club and spades.

∴ No. of favourable outcomes = 6 × 4 = 24.

P(getting a card with number between 2 and 9)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{24}{52} = \dfrac{6}{13}$.

Hence, the probability of drawing a card with number between 2 and 9 = $\dfrac{6}{13}$.