72 ELEMENTARY PROBABILITY Ch. 3 



average winnings per game and is, therefore, the amount you could 

 pay to play this game and expect to break even. 



The preceding ideas and methods can be generalized and sym- 

 bolized in the following manner. Let all the single events possible 

 under a specified set of conditions be grouped into s mutually ex- 

 clusive classes of events. Let Xt be the reward, loss, or in general 

 the "value" of the occurrence of an event in the ith class; and let 

 Pi be the probability that an event in the ith class will occur on any 

 designated future trial under the stipulated conditions. Finally, let 

 E(x) stand for the total mathematical expectation under the given 

 conditions. It follows from the reasoning outlined above that 



s 



(3.42) E(x) = p lXl + p 2 -r 2 + • • ■ + p*x s = £ ( PiXi ). 



i=l 



PROBLEMS 



1. Suppose that you are 20 years of age and that you are to inherit $10,000 

 at the age of 30 if you are alive then. What is the expected value of this in- 

 heritance if you have a probability of .92 of living to be 30 years of age? (This 

 probability is derived from the American Experience Mortality Table.) 



2. It is approximately true that brown and blue eye colors are inherited in 

 a manner similar to that explained for the A-B blood groups. If b/b = blue 

 eye color and either B/b or B/B = brown eye color, what is the expected num- 

 ber of blue-eyed children among 500 from parents who are B/b and b/b, re- 

 spectively? Ans. 250. 



3. Answer the same question as in problem 2 for parents who are both B/b. 



4. If in each three-month period 1 car in 20 of the type which you drive has 

 an accident costing an average of $75 for repairs, how much insurance against 

 such a loss should you pay each quarter if you allow the company 15 per cent 

 beyond mathematical expectation for handling the business, and if you ignore 

 interest on your money? Ans. $4.31. 



5. How much would one be justified mathematically in wagering against one 

 dollar that on 10 throws of two unbiased dice a sum of 7 will appear less than 

 3 times? 



6. Suppose that you have the choice of receiving $10,000 at age 65 if you are 

 alive then, or of taking a cash payment now. From a purely mathematical 

 point of view and ignoring interest on money, what should the size of the pay- 

 ment be if your probability of living to be 65 is .56? Ans. $5600. 



7. Suppose that a concession at a fair offers a 50-cent prize if you pay 10 

 cents for 3 throws and knock down all of a stack of milk bottles on the 3 

 throws. Suppose also that you have 1 chance in 10 to knock down the bottles. 

 If the operator of the concession has to pay $75 per day for the privilege of 

 doing business there, how many customers must he have per day in order that 

 he can expect (mathematically) to make some money? 



8. Suppose that a person who is 40 years of age is to receive $1000 on each of 

 his sixtieth, sixty-first, and sixty-second birthdays if he is alive to receive them. 

 Also suppose that interest on money is to be ignored. Given that his proba- 



