Question

Three cards are drawn from a pack of 52 cards. Find the chance that 

(a) two are queen cards and one is an ace card. 

(b) at least one is a diamond card. 

(c) all are from the same suit. 

(d) they are a king, a queen, and a jack.

Answer 1

3 cards can be drawn from a pack of 52 cards in ⁵²C₃ ways.

∴ n(S) = ⁵²C₃

(a) Let event A: Out of the three cards drawn, 2 are queens and 1 is an ace card. 

There are 4 queens and 4 aces in a pack of 52 cards. 

∴ 2 queens can be drawn from 4 queens in ⁴C₁ ways and 1 ace can be drawn out of 4 aces in ⁴C₁ ways.

(b) Let event B: Out of the three cards drawn, at least one is a diamond. 

∴ B’ is the event that all 3 cards drawn are non-diamond cards. 

In a pack of 52 cards, there are 39 non-diamond cards.

∴ 3 non-diamond cards can be drawn in ³⁹C₃ ways.

(c) Let event C: All the cards drawn are from the same suit. A pack of 52 cards consists of 4 suits each containing 13 cards. 

∴ 3 cards can be drawn from the same suit in ¹³C₃ ways.

(d) Let event D: The cards drawn are a king, a queen, and a jack.

There are 4 kings, 4 queens and 4 jacks in a pack of 52 cards. 

∴ 1 king can be drawn from 4 kings in ⁴C₁ways, 1 queen can be drawn from 4 queens in ⁴C₁ways and 1 jack can be drawn from 4 jacks in ⁴C₁