Poisson Distribution

To answer by the book: 6.

Explanation

Given r = 3, we first want to know the maximum likely number of total customers during the final half hour. The Poisson Table gives values up to p(12), which is 0.0001, but this and its neighbors are so unlikely as to be negligible. If we add probabilities, beginning with the highest given value and moving up the column, we will sooner or later find a point where the cumulative total is just short of 0.1000; that is, where there is a 10% chance (or better) that that many gold-bearing customers will arrive during the final half hour. From the table, that point is reached at p(6), the cumulative total being then 0.0839. That is, there is a 92.61% chance that 6 or fewer customers will visit during the last half hour Friday. That will take in the decently likely cases. Since there is no guarantee that the customers will be evenly spaced, the bank in its prudence may decide to assign 6 employees to turn teller for that final half hour.

Comment

That is the answer by the book. Good for you if you got that answer. But in real life, the bank will not have six tellers waiting at six windows for what on average will only be three customers. It's intimidating to the customers, if the truth be told, and it's wasteful of staff. Instead, there will be, say, two tellers at the windows, and four employees doing other work, but reassignable to windows as need may arise. The senior teller, besides tending to her own window, is in charge of giving that signal. The work week will be laid out so that the signal can be answered if given, without creating a crisis with things that have to be done before closing time Friday. Laying it out that way is part of the experiment design, or as we may prefer to say, the bank's operational plan.

A look at the detailed records for previous Fridays, the records from which the rate r = 3 was derived, may well give better information than the Poisson Table about how high the Friday peaks are likely to run. We only use the Poisson table when we have to guess that peak number from general probability. Poisson probability is the right kind of probability to make that guess with; no question about it. But empirical experience may give a better guideline. It seems somehow perverse to compress empirical experience into one number, the rate r, and then guess it back again from general principles, however theoretically appropriate those principles may be.