To determine who should be the first dealer in a card game, one card is dealt to each of five players. The player with the card of the highest denomination gets to deal first.

Question

a) How many different results are possible when dealing to the five players?

b) In how many ways could all players receive cards of different denominations?

c) What is the probability that four players receive cards of the same denomination?

d) How would the solution to part c) change if players each chose a card from a full deck instead of being dealt one?

Answer 1

52 card deck

if order doesn't matter

when dealt one card to each of 5 players

possibilities are

52C5 = 52!/47!5! = 52(51)(50)(49)(48)/5(4)(3)(2)

=2.598,960

= over 2 1/2 million different hands possible

b) Probability each player will receive different denominations= number of ways of getting different denominations divided by 2598960 = 1- probability in part c) = 2598947/2598960 = close to 100%

c) Probability each player will receive the same denomination = number of ways of getting the same denomination divided by 2598960 Number of ways of getting the same denomination is 13, either all aces, all kings, all queens, all jacks, all 10s, all 9s, .... all 3's or all 2's. Probability = 13/2598960 = close to zero

= about 5 x 10^-6 = .000005 = .0005%

or 13(1/13)(12/51)(11/50)(10/49) =about .01056? ?

d) same as c) but if each player took a card from a full deck? rather than all taking from the same deck

(1/13)^4 = 1/28,568 =about 3.5 x 10^-5 = .000035 = .0035% which is better odds than .0005%, about 7 times better odds