Page 115 Ques 20

20 A six-card hand is dealt from an ordinary deck of cards. Find the probability that:

All six cards are hearts.

We have 52 cards choose 6. There are 13 heart cards to choose 6.

P(all 6 cards are hearts) = \(\dfrac {13 \choose 6} {52 \choose 6}\) = \(\dfrac {1716} {20358520}\) = 0.000084

There are three aces, two kings, and one queen.

We have 52 cards choose 6. There are 4 aces cards to choose 3, 4 kings to choose 2 and 4 queens to choose 1.

P(three aces, two kings, and one queen) = \(\dfrac {{4 \choose 3} \times {4 \choose 2} \times {4 \choose 1}} {52 \choose 6}\) = (4 x 6 x 4) / 20358520 = 0.0000047

There are three cards of one suit and three of another suit.

There are 4 suites to choose 2, 13 cards in one suit to choose 3 and 13 cards in another suit to choose 3.

P(three cards of one suit and three of another suit) = \({4 \choose 2} \times \dfrac {{13 \choose 3} \times {13 \choose 3}} {52 \choose 6}\) = 6 x (286 x 286 / 20358520) = 0.0241