Exercise
Mr. Wimply Dimple, one of London's most prestigious watch
makers, has come to Sherlock Holmes in a panic, having discovered that someone has been producing and selling crude counterfeits of his best selling watch. The 16 counterfeits so far discovered bear stamped numbers, all of which fall between 1 and 56, and Dimple is anxious to know the extent of the forger's work. All present agree that it seems reasonable to assume that the counterfeits thus far produced bear consecutive numbers from 1 to whatever the total number is.
“Chin up, Dimple,” opines Dr.\ Watson. “I shouldn't worry overly much if I were you;
the Maximum Likelihood Principle, which estimates the total
number as precisely that which gives the highest probability for the series of numbers found,
suggests that we guess 56 itself as the total. Thus, your forgers are not a big operation, and we
shall have them safely behind bars before your business suffers significantly.”
“Stuff, nonsense, and bother your fancy principles, Watson,” counters Holmes.
“Anyone can see that, of course, there must be quite a few more than 56 watches---why
the odds of our having discovered precisely the highest numbered watch made are
laughably negligible. A much better guess would be twice 56.”
- Show that Watson is correct that the Maximum Likelihood Principle gives 56.
- Write a computer program to compare Holmes's and Watson's guessing strategies as follows: fix a total [math]N[/math] and choose 16 integers randomly between 1 and [math]N[/math]. Let [math]m[/math] denote the largest of these. Then Watson's guess for [math]N[/math] is [math]m[/math], while Holmes's is [math]2m[/math]. See which of these is closer to [math]N[/math]. Repeat this experiment (with [math]N[/math] still fixed) a hundred or more times, and determine the proportion of times that each comes closer. Whose seems to be the better strategy?