66 ELEMENTARY PROBABILITY Ch. 3 



Problem 3.22. How many different (as to cards held) 5-card poker hands are 

 possible from the usual 52-card deck? 



In view of the fact that the order in which the cards were dealt 

 does not affect the actual cards held, this is a problem in numbers 

 of combinations of 52 objects taken 5 at a time; hence the number 

 of poker hands is C 52 , 5 = (52 1)/ (5 147!) = 53,040 after common fac- 

 tors in numerator and denominator are divided out and the remain- 

 ing factors multiplied together. 



Problem 3.23. What is the probability that 5 cards dealt from a well-shuffled 

 poker deck will all be spades? 



Two numbers need to be determined before formula 3.11 can be 

 applied: (1) the total number of 5-card hands which are all spades, 

 and (2) the total number of 5-card hands of any sort which possibly 

 could be dealt from the deck. In view of the fact that the order in 

 which the cards were dealt is unimportant, this is a matter of finding 

 numbers of combinations, namely, C 13< 5 and C 52 , 5 . Therefore, the 

 required probability is 



P(all spades) = C 13 , 5 /C 52 , 5 = -0005, or 1 chance in 2000. 



Problem 3.24. What is the probability that 5 cards dealt from a well-shuffled 

 poker deck will include exactly 3 aces? 



Three aces can be chosen from among the 4 available in C 4> 3 or 

 4 ways. Likewise, C 48> 2 = 1128 is the number of different pairs of 

 cards which do not include any aces. All possible 5-card hands with 

 exactly 3 aces must necessarily be the same as all the possible ways 

 to put some 3 aces with one of the 1128 pairs of cards which are not 

 aces; hence there must be 4(1128) = 4512 different 5-card hands 

 which include exactly 3 aces. Therefore, the probability of being 

 dealt such a hand is 



P(exactly 3 aces, 2 non-aces) = 4512/C 52 , 5 



= .0016, or 1 chance in 625. 



PROBLEMS 



1. In how many ways, which differ as regards the persons in particular chairs, 

 can 4 men and 4 women be seated around a dinner table, with men and women 

 seated alternately? 



2. Suppose that there are 10 persons in a room, and that they have the fol- 

 lowing blood types: 1 is AB, 3 are A, 2 are B, and 4 have type O blood. If 2 



