Question

Two cards are drawn from a well shuffled pace of 52 cards. Find the probability that

(1) both the cards are of different colour
(2) both the cards are face cards
(3) one of the two cards is a king.

Answer 1

Total number of primary outcomes of drawing two cards from a well shuffled pack of 52 cards is

n = ⁵²C₂ = \(\frac{52×51}{2×1} =1326\)

(1) A = Event that both the cards drawn are of different colours, i.e., one card is of black colour and one is of red colour.

In a pack of 52 cards, 26 cards are black and 26 cards are red.

∴ Favourable outcomes for the event A is

m = ²⁶C₁ × ²⁶C₁ = 26 × 26 = 676

Hence, P(A) = \(\frac{m}{n} = \frac{676}{1326} = \frac{26}{51}\)

(2) B = Event that both the cards are face cards. In a pack of 52 cards, 12 cards are face cards.

∴ Favourable outcomes for the event B is

m = ¹²C₂ = \(\frac{12×11}{2×1} = 66.\)

Hence, p(B) = \(\frac{m}{n} =  \frac{66}{1326} = \frac{11}{221}\)

(3) C = Event that one of the two cards is a king. In a pack of 52 cards, 4 cards are of king and other cards are 48.

∴ Favourable outcomes for the event C is

m = ⁴C₁ × ⁴⁸C₁ = 4 × 48 = 192

Hence, P(C) =\(\frac{ m}{n} = \frac{192}{1326} = \frac{32}{221}\)