Two cards are drawn at random from a well-shuffled pack of 52 cards. W

1.	Two cards are drawn at random from a well-shuffled pack of 52 cards. What is the probability that either both are black or both are kings ?(a) \(\frac{55}{112}\)(b) \(\frac{55}{221}\)(c) \(\frac{33}{221}\)(d) \(\frac{33}{112}\)
Answer» (b) \(\frac{55}{221}\) S : Drawing 2 cards out of 52 cards ⇒ n(S) = ⁵²C₂ = \(\frac{\|\underline{52}}{\|\underline{52}\|\underline2}\) = \(\frac{52\times51}{2}\) = 1326 A : Event of drawing 2 black cards out of 26 black cards ⇒ n(A) = ²⁶C₂ = \(\frac{26\times25}{2}\) = 325 B : Event of drawing 2 kings out of 4 kings ⇒ n(B) = ⁴C₂ = \(\frac{\|\underline{4}}{\|\underline{2}\|\underline2}\) = 6 ⇒ A ∩ B : Event of drawing 2 black kings ⇒ n(A ∩ B) = ²C₂ = 1 ∴ P(A) = \(\frac{325}{1326}\), P(B) = \(\frac{6}{1326}\), P(A ∩ B) = \(\frac{1}{1326}\) ∴ P(Both cards are black or both kings) = P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = \(\frac{325}{1326}\) + \(\frac{6}{1326}\) - \(\frac{1}{1326}\) = \(\frac{330}{1326}\) = \(\frac{55}{221}\).

Two cards are drawn at random from a well-shuffled pack of 52 cards. What is the probability that either both are black or both are kings ?(a) \(\frac{55}{112}\)(b) \(\frac{55}{221}\)(c) \(\frac{33}{221}\)(d) \(\frac{33}{112}\)

Answer»

(b) \(\frac{55}{221}\)

S : Drawing 2 cards out of 52 cards

⇒ n(S) = ⁵²C₂ = \(\frac{|\underline{52}}{|\underline{52}|\underline2}\) = \(\frac{52\times51}{2}\) = 1326

A : Event of drawing 2 black cards out of 26 black cards

⇒ n(A) = ²⁶C₂ = \(\frac{26\times25}{2}\) = 325

B : Event of drawing 2 kings out of 4 kings

⇒ n(B) = ⁴C₂ = \(\frac{|\underline{4}}{|\underline{2}|\underline2}\) = 6

⇒ A ∩ B : Event of drawing 2 black kings

⇒ n(A ∩ B) = ²C₂ = 1

∴ P(A) = \(\frac{325}{1326}\), P(B) = \(\frac{6}{1326}\), P(A ∩ B) = \(\frac{1}{1326}\)

∴ P(Both cards are black or both kings)

= P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

= \(\frac{325}{1326}\) + \(\frac{6}{1326}\) - \(\frac{1}{1326}\) = \(\frac{330}{1326}\) = \(\frac{55}{221}\).