70 ELEMENTARY PROBABILITY Ch. 3 



8. If a pair of unbiased dice is to be thrown 6 times in succession, what is the 

 probability that exactly 3 sevens will be thrown? What would you think the 

 most likely number of sevens would be? Ans. .054. 



9. If a certain manufacturing process is producing machine parts of which 10 

 per cent have some serious defect, what is the probability that all of the 10 

 parts chosen at random will be acceptable (that is, have no serious defect)? 

 How many would you have to take in the sample before the probability of all 

 being acceptable will be no greater than .05? 



10. Graph j(p) = (1 — p) 10 , and relate this graph to problems like problem 9. 



3.4 MATHEMATICAL EXPECTATION 



The discussions earlier in this chapter have involved the occur- 

 rences of chance events as a result of what have been termed 

 "trials" under specified conditions. The outcome of a trial is de- 

 scribed in one of two general ways: (a) Something happens a certain 

 number of times on a specified number of trials, or (b) we simply 

 note whether or not an event E has, or has not, occurred and asso- 

 ciate with that occurrence some value, say a financial loss, as in in- 

 surance. With either type of situation it may be important to be 

 able to predict what will be the average outcome of trials under the 

 stated conditions, over the long run of experience. For example, an 

 insurance company needs to know what amounts it should expect 

 to have to pay out in death benefits during a particular period of 

 time, one year, for instance. 



In case a, the prediction needed is to be presented in the form of an 

 expected number of occurrences of an event E on a set of n future 

 trials. A formula for this expected number can be justified heuristi- 

 cally as follows. If the probability of E is p, the p is just the fraction 

 of the time that E should occur over many trials. Hence, if there are 

 to be n trials, it is reasonable to say that the expected number of 

 occurrences of E on n trials is 



(3.41) Expected number = E(r) = p-(n). 



Problem 3.41. If 6 unbiased coins are to be tossed simultaneously, what is the 

 expected number of heads? 



In this circumstance p = 1/2 and n = 6; hence the expected (or 

 long-run average) number of heads is E (r) = (1/2) (6) = 3. Actu- 

 ally, our intuition would lead to the same conclusion. 



Problem 3.42. Suppose that an insurance company has insured 50,000 persons 

 who are each 30 years old, and that records from past experience show that 

 6/1000 of such persons die before reaching the age of 31. What is the expected 

 number of deaths during the first year of the insurance contract? 



