Sec. 3.3 REPEATED TRIALS 69 



(1/2 -f- 1/2) n and (q + p) re , in which q = 1 — p. To see these 

 generalizations, consider the following binomials expansions: 



(1/2 + 1/2) 2 = l(l/2)°(l/2) 2 + 2(l/2) 1 (l/2) 1 + l(l/2) 2 (l/2)°, 



= C 2 , (l/2)°(l/2) 2 + C 2 , Aimwn? 



+ C 2 , 2 (l/2) 2 (l/2)°, 



= P(0H, 2T) + P(1H, IT) + P(2H, OT). 



That is, the successive terms of the expansion of (1/2 -f- 1/2 ) 2 are 

 given by formula 3.31 if r = 0, 1, and 2, successively; and those 

 three terms give the probabilities for the three possible classes of 

 events in terms of the number of heads appearing. The generaliza- 

 tions for 3, 4, . . ., or n tosses should be apparent. For the more 

 general situation in which the probability of the occurrence of an 

 event E is constantly p under repeated trials, 



(9 + V? = Hp)°(q) 2 + 2(p) 1 (q) 1 + l(p) 2 (q)°, 



= P(0 E'b, 2 not-£'s) + P(l E, 1 not-E) 



+ P(2 E'b, note's) ; 



and again it should be apparent that these successive terms corre- 

 spond to formula 3.32 for r = 0, 1, and 2, successively. 



PROBLEMS 



1. What is the probability that if 6 unbiased pennies are tossed simulta- 

 neously, exactly 3 heads will appear? 



2. What is the probability that at least 3 heads will appear under the condi- 

 tions of problem 1? Ans. 21/32. 



3. If one parent is Rhrh and AO, and the other parent is rhrh and BO, what 

 is the probability that both their first two children will be Rh— and AB? 



4. Suppose that a sample of 100 bolts is taken from a very large batch which 

 contains exactly one-half of 1 per cent of unacceptable bolts. What is the 

 probability that at least 2 bolts in the sample will be unacceptable? Ans. .09. 



5. If 5 bolts among the 100 in the sample of problem 4 are found to be un- 

 acceptable products, what would you conclude about the hypothesis that only 

 one-half of 1 per cent were faulty in the whole batch? Give reasons. 



6. Write out the series for (x + y) 4 and show that the coefficients are num- 

 bers of combinations, C i r , with r = to 4. 



7. Suppose that the teams listed on a football parlay card are so handicapped 

 that you actually have a 50-50 chance on each team you pick. What is the 

 probability that you will pick exactly 9 winners out of 10? Would this proba- 

 bility justify odds of 25 to 1 for this accomplishment? What about odds of 

 250 to 1 for getting 10 out of 10 correct? 



