Eight players play a card game with a 52 card deck The deck

Eight players play a card game with a 52 card deck. The deck is shuffled and each player is dealt four cards.

a.) What is the probability that player A gets all four kings?

b.) What is the probability that some player gets all four kings?

c.) What is the probability that player A gets four cards of a kind? (That is, four aces, four kings, four queens, ... four aces.)

d.) What is the probability that some player gets four cards of a kind? Why is it a lot harder to go from answer c. to answer d. than it was to go from answer a. to answer b.?

Solution

8 players, each getting 4 cards

So, 32 cards will be dealt in all

To get all the four kings, one must ensure that the deck has all the four kings,

the sample would be number of ways of choosing 28 cards from 48 and the four kings = ⁴⁸C₂₈

The number of ways in which the four kings can be given to a specific player = 1

Thus, required probability

= 1 /  ⁴⁸C₂₈

b) If one of the 8 players is to get the four kings, the probability just gets multiplied by 8

Hence the required probability = 8 /  ⁴⁸C₂₈

c) This part gets more complex.

The denominator here = ⁵²C₃₂

The problem is that there might be more than one four of kind in the 32 cards thus selected

Also, it is difficult to determine which card value is as four of a kind (aces, kings, queens, . , etc...)

Going from C to D is tougher because of the number of permutations increase in number with the changes in the card value as well as the player who gets four of a kind