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

(i) '2' of spades

(ii) a jack

(iii) a king of red colour

(iv) a card of diamond

(v) a king or a queen

(vi) a non-face card

(vii) a black face card

(viii) a black card

(ix) a non-ace

(x) non-face card of black colour

(xi) neither a spade nor a jack

(xii) neither a heart nor a red king.

Well-shuffling ensures equally likely outcomes.

Total number of outcomes = 52.

(i) There is only one 2 of spades in the whole pack.

∴ P('2' of spades) = $\dfrac{1}{52}$.

Hence, the probability of drawing 2 of spades = $\dfrac{1}{52}$.

(ii) There are 4 jacks, one of each suit.

∴ The number of favourable outcomes to the event 'a jack' = 4.

∴ P(a jack) = $\dfrac{4}{52} = \dfrac{1}{13}$.

Hence, the probability of drawing a jack = $\dfrac{1}{13}$.

(iii) There are two king of red colour, one of hearts and one of diamond.

∴ The number of favourable outcomes to the event 'king of red colour' = 2.

∴ P(king of red colour) = $\dfrac{2}{52} = \dfrac{1}{26}.$

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

(iv) There are 13 cards of diamond suit.

∴ The number of favourable outcomes to the event 'a card of diamond' = 13.

∴ P(a card of diamond) = $\dfrac{13}{52} = \dfrac{1}{4}$.

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

(v) There are 8 king and queen cards, 2 in each suit.

∴ The number of favourable outcomes to the event 'a king or queen' = 8.

∴ P(a king or queen) = $\dfrac{8}{52} = \dfrac{2}{13}$.

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

(vi) There are 12 face cards.

∴ No. of non-face cards = 52 - 12 = 40.

∴ The number of favourable outcomes to the event 'a non-face card' = 40.

∴ P(a non-face card) = $\dfrac{40}{52} = \dfrac{10}{13}$.

Hence, the probability of drawing a non-face card = $\dfrac{10}{13}$.

(vii) Since 2 suits are of black colour and each suit has 3 face cards.

∴ No. of black face cards = 2 × 3 = 6.

∴ The number of favourable outcomes to the event 'a black face card' = 6.

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

Hence, the probability of drawing a black face card = $\dfrac{3}{26}$.

(viii) There are 2 suits of black cards.

∴ No. of black cards = 26.

∴ The number of favourable outcomes to the event 'a black card' = 26.

∴ P(a black card) = $\dfrac{26}{52} = \dfrac{1}{2}$.

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

(ix) There are 4 ace cards, one in each suit.

∴ No. of non-ace cards = 52 - 4 = 48.

∴ The number of favourable outcomes to the event 'a non-ace card' = 48.

∴ P(a non-ace card) = $\dfrac{48}{52} = \dfrac{12}{13}$.

Hence, the probability of drawing a non-ace card = $\dfrac{12}{13}$.

(x) There are 3 face cards in each suit and 2 suits of black colour.

Hence, no. of face cards of black colour = 6.

∴ No. of non-ace black cards = 26 - 6 = 20.

∴ The number of favourable outcomes to the event 'a non-face card of black colour' = 20.

∴ P(a non-face black card) = $\dfrac{20}{52} = \dfrac{5}{13}$.

Hence, the probability of drawing a non-face black card = $\dfrac{5}{13}$.

(xi) There are 13 spade cards and each suit has 1 jack.

So, the other 3 suits apart from spade has 3 jacks.

∴ Total no. of spade and jack cards = 13 + 3 = 16.

Hence, no. of cards other than spade and jack = 52 - 16 = 36.

∴ The number of favourable outcomes to the event 'neither a spade nor a jack' = 36.

∴ P(neither a spade nor a jack) = $\dfrac{36}{52} = \dfrac{9}{13}$.

Hence, the probability of drawing neither a spade nor a jack = $\dfrac{9}{13}$.

(xii) There are 13 heart cards and 2 suits of red colour.

Since, one king of red colour is already included in hearts hence only one red king is more.

∴ Total no. of heart and red king cards = 13 + 1 = 14.

Hence, no. of cards other than heart and red king cards = 52 - 14 = 38.

∴ The number of favourable outcomes to the event 'neither heart nor a red king' = 38.

∴ P(neither a heart nor a red king) = $\dfrac{38}{52} = \dfrac{19}{26}$.

Hence, the probability of drawing neither a heart nor a red king = $\dfrac{19}{26}$.