Max the bookie is trying to decide how many telephones to install in his new bookmaking operation…

Max the bookie is trying to decide how many
telephones to install in his new bookmaking operation. Because of heavy police
activity, he cannot increase or decrease the number of telephones once he sets
up his operation. He has narrowed the possible choices to three: he can install
25, 50 or 100 telephones. His profit for one year (the usual length of time he
can remain in business before the police close him down) depends on the average
number of calls he receives. (Calls occur randomly and independently of one
another.) After some deliberation he concludes that the average number of calls
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Max the bookie is trying to decide how many
telephones to install in his new bookmaking operation. Because of heavy police
activity, he cannot increase or decrease the number of telephones once he sets
up his operation. He has narrowed the possible choices to three: he can install
25, 50 or 100 telephones. His profit for one year (the usual length of time he
can remain in business before the police close him down) depends on the average
number of calls he receives. (Calls occur randomly and independently of one
another.) After some deliberation he concludes that the average number of calls
per minute can be 0.5, 1.0 or 1.5 with probabilities of 0.50, 0.25 and 0.25,
respectively. Max then produces the payoffs given in the following table. Max’s
assistant, Lefty, points out that Max may be able to get more information by
observing a competitor’s similar operation. However, he will only be able to
watch for 10 minutes, and doing so will cost him $4000. Max determines that if
he counts fewer than eight calls, that would be a low number; at least eight
but fewer than 17 would be a medium number; and at least 17 would be a large
number of calls. Max also decides that, if the experiment is run, he will only
record whether there is a small, medium or large number of calls. Help Max by
performing a preposterous analysis to determine whether the sample should be taken.
Conclude by specifying clearly what the optimal strategy is. (Hint: The number
of telephone calls is Poisson-distributed.)

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