A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that it is :

(i) an ace

(ii) a jack of hearts

(iii) a three of clubs or a six of diamonds

(iv) a heart

(v) any suit except heart

(vi) a ten or a spade

(vii) neither a four nor a club

(viii) a picture card

(ix) a spade or a picture card.

Total cards = 52

∴ No. of possible outcomes = 52

(i) There are 4 aces in a deck (1 of each suit).

∴ No. of favourable outcomes = 4

P(drawing an ace) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{4}{52} = \dfrac{1}{13}$.

Hence, probability of drawing an ace = $\dfrac{1}{13}$.

(ii) There is only one jack of hearts.

∴ No. of favourable outcomes = 1

P(drawing a jack of hearts) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{1}{52}$.

Hence, probability of drawing a jack of hearts = $\dfrac{1}{52}$.

(iii) There is one three of clubs and one six of diamonds.

∴ No. of favourable outcomes = 2

P(drawing a three of clubs or a six of diamonds)

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

Hence, probability of drawing a three of clubs or a six of diamonds = $\dfrac{1}{26}$.

(iv) There are 13 hearts in a deck.

∴ No. of favourable outcomes = 13

P(drawing a heart) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{13}{52} = \dfrac{1}{4}$.

Hence, probability of drawing a heart = $\dfrac{1}{4}$.

(v) There are 39 other cards except heart.

∴ No. of favourable outcomes = 39

P(drawing any suit except heart) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{39}{52} = \dfrac{3}{4}$.

Hence, probability of drawing any suit except heart = $\dfrac{3}{4}$.

(vi) There are 13 spade cards and 3 other 10's (1 of each suit except spade)

∴ No. of favourable outcomes = 16

P(drawing a ten or a spade)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{16}{52} = \dfrac{4}{13}$.

Hence, probability of drawing a ten or a spade = $\dfrac{4}{13}$.

(vii) There are 13 club cards and 3 other four cards (1 of each suit apart from club)

Total cards = 16

Left cards = 52 - 16 = 36.

∴ No. of favourable outcomes = 36

P(drawing neither a club nor a four)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{36}{52} = \dfrac{9}{13}$.

Hence, probability of drawing neither a club nor 4 = $\dfrac{9}{13}$.

(viii) Jack, King and Queen are considered as picture cards.

There are 4 jacks, 4 queens and 4 kings

∴ No. of favourable outcomes = 12

P(drawing a picture card)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{12}{52} = \dfrac{3}{13}$.

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

(ix) No. of spade cards = 13

Jack, king and queen except that of spade are 3 each.

Total cards = 13 + 3 + 3 + 3 = 22.

∴ No. of favourable outcomes = 22

P(drawing a spade or a picture card)

= $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{22}{52} = \dfrac{11}{26}$.

Hence, probability of drawing a spade or a picture card = $\dfrac{11}{26}$.