Sec. 3.3 REPEATED TRIALS 67 



are chosen at random what is the probability that they will have the same type 

 of blood? Ans. 2/9. 



3. Suppose that a baseball team has 4 men who can bat in any of the first 

 3 positions, 5 who can bat in any of the fourth, fifth, and sixth positions, and 

 7 who can bat in any of the last three positions. How many possible batting 

 orders are there? 



4. Assume that 7 insecticides are to be tested as to their effectiveness in 

 killing house flies. If each spray is to be tested against every other spray once 

 in a separate test, how many tests will this require? Ans. 21. 



5. Suppose that a housewife buys 3 cans of peaches, 6 cans of apricots, and 4 

 cans of pears; and suppose that her child tears off all the labels on the cans. 

 If the housewife needs 2 cans of fruit for dinner, what is the probability that 

 the first 2 cans chosen will contain the same kind of fruit? 



6. How many 13-card bridge hands are there with no card higher than 8? 



Ans. 37,442,160. 



7. If 7 unbiased coins are flipped simultaneously, how many single events are 

 there in the class: 3 heads, 4 tails? 



8. Compare the coefficients of (x -\- y) 5 with C 5 5 , C 5 4 , C 5 3 , C 5 9 , C 5 ,, 

 and C 5 given that 0! = 1. 



9. What is the probability that 5 cards dealt from a well-shuffled poker deck 

 will include 3 queens and 2 aces? Three queens and at least 1 ace? 



10. What is the probability that 13 cards dealt from a well-shuffled bridge 

 deck will include exactly 8 honor cards (honor cards are 10, J, Q, K, and Ace)? 



Ans. .040. 



11. In how many ways can 6 boxers be paired off for 3 bouts being held 

 simultaneously? 



3.3 REPEATED TRIALS UNDER SPECIFIED 

 CONDITIONS 



Situations involving the numbers of occurrences and non-occur- 

 rences of an event E on repeated trials under the same original con- 

 ditions are of particular interest in statistical analysis. The prin- 

 ciples involved will be seen to be important to the study of frequency 

 distributions, and to sampling studies. 



The probability problems created when trials are repeated under 

 fixed conditions can be illustrated by means of mathematical models 

 of these problems. Suppose that a coin is flipped n times and the 

 number of heads noted. On such a set of repeated trials any number 

 of heads is possible from to n, that is, there are (n + 1) possible 

 classes of event: heads, n tails; 1 head, (n — 1) tails; 2 heads, 

 (n — 2) tails; . . . ; n heads, tails. Each class of events includes 

 some number of single events (if the coin is unbiased) from 1 to 

 whatever the maximum size of C n , r is for the given n. For example, 



