A card is drawn from a well-shuffled pack of 52 cards. Find the probability that the card drawn is :

(i) a spade.

(ii) a red card.

(iii) a face card.

(iv) 5 of heart or diamond.

(v) Jack or queen.

(vi) ace and king.

(vii) a red and a king.

(viii) a red or a king.

There are 52 cards in a deck which are divided into 4 suits of 13 cards each.

∴ No. of possible outcomes = 52.

(i) There are 13 spades in a deck of playing cards.

∴ No. of favourable outcomes = 13.

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

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

(ii) There are 26 red cards (13 hearts and 13 diamonds).

∴ No. of favourable outcomes = 26.

P(drawing a red card) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{26}{52} = \dfrac{1}{2}$.

Hence, the probability of drawing a red card = $\dfrac{1}{2}$.

(iii) There are 12 face cards (4 kings, 4 queens, 4 jacks) in a deck.

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

(iv) There are 2 cards one of each heart and diamond with no. 5.

∴ No. of favourable outcomes = 2.

P(drawing a 5 of heart or diamond) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{2}{52} = \dfrac{1}{26}$.

Hence, the probability of drawing a 5 of heart or diamond = $\dfrac{1}{26}$.

(v) There are 4 jacks and 4 queens in a deck.

∴ No. of favourable outcomes = 8.

P(drawing a jack or queen) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{8}{52} = \dfrac{2}{13}$.

Hence, the probability of drawing a jack or queen = $\dfrac{2}{13}$.

(vi) An ace and a king cannot be in a single card.

∴ No. of favourable outcomes = 0.

P(drawing an ace and a king) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}}$ = 0.

Hence, the probability of drawing an ace and a king = 0.

(vii) The king of diamond and heart is red in colour.

∴ No. of favourable outcomes = 2.

P(drawing a red and a king) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{2}{52} = \dfrac{1}{26}$.

Hence, the probability of drawing a red and a king = $\dfrac{1}{26}$.

(viii) There are 26 red (13 hearts + 13 diamonds) and 1 king of each (club and spade).

∴ No. of favourable outcomes = 26 + 1 + 1 = 28.

P(drawing a red or a king) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{28}{52} = \dfrac{7}{13}$.

Hence, the probability of drawing a red or a king = $\dfrac{7}{13}$.