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

(i) a red face card

(ii) neither a club nor a spade

(iii) neither an ace nor a king of red colour

(iv) neither a red card nor a queen

(v) neither a red card nor a black king.

Well shuffling ensures equally likely outcomes.

Total number of outcomes = 52.

(i) Each suit has one king, one queen and one jack, and there are two suits of red colour.

∴ There are 2 kings, 2 queens and 2 jacks.

∴ The number of red face cards = 6.

∴ P(a red face card) = $\dfrac{6}{52} = \dfrac{3}{26}.$

Hence, the probability that a card drawn is a red face card = $\dfrac{3}{26}$.

(ii) There are 13 clubs and 13 spades i.e. total = 26 cards.

No. of cards left other than spades and clubs = 52 - 26 = 26.

P(neither a club nor spade) = $\dfrac{26}{52} = \dfrac{1}{2}.$

Hence, the probability that a card drawn is neither a club nor spade card = $\dfrac{1}{2}$.

(iii) There are 2 kings of red colour, one of heart and one of diamond.

There are 4 aces, one of each suit.

No. of cards other than ace and king of red colour = 52 - 4 - 2 = 46.

P(neither an ace nor a king of red colour) = $\dfrac{46}{52} = \dfrac{23}{26}.$

Hence, the probability that a card drawn is neither an ace nor a king of red colour = $\dfrac{23}{26}$.

(iv) There are 26 red cards, 13 of hearts and 13 of diamonds.

There are 4 queens, one of each suit but since red queens are included in red cards hence, queens left = 2.

Total no. of red cards and queen = 26 + 2 = 28.

No. of cards other than queen and red cards = 52 - 28 = 24.

P(neither a red card nor queen) = $\dfrac{24}{52} = \dfrac{6}{13}.$

Hence, the probability that a card drawn is neither a red card nor queen = $\dfrac{6}{13}$

(v) There are 26 red cards, 13 of hearts and 13 of diamonds.

There are 2 black kings, one of club and one of spade.

Total no. of red cards and black kings = 26 + 2 = 28.

No. of cards other than red cards and black kings = 52 - 28 = 24.

P(neither a red card nor black king) = $\dfrac{24}{52} = \dfrac{6}{13}.$

Hence, the probability that a card drawn is neither a red card nor black king = $\dfrac{6}{13}$