Your country is at war, and an enemy plane has crashed on your territory. It bears the number 60, and a spy has told you that the aircraft are numbered serially. Can you make a guess about the total number of aircraft the enemy has produced?
Solution: This problem is one of those published by Frederick Mosteller, the Harvard statistician, in “Fifty Challenging Problems in Probability.” Mosteller notes that there is no single correct answer. He describes several approaches, and their statistical properties:
Strategy 1: You want to be right, on average. In other words, you seek a procedure that, in many such situations, will yield estimates that average out to be the right answer. Mosteller likens the aircraft serial number to the random selection of a point on a ruler. If you repeatedly select a point on a ruler, at random, the midpoint of the ruler is the average of all such selections, and the best guess about where a given selection lies. According to this strategy of “being correct on average,” if 60 is the “ruler’s” midpoint, then 120 would be the maximum. So 120 would be your guess about how many aircraft were produced.
Strategy 2: You want to have the maximum probability that your guess will be correct (exactly). This approach illustrates the “maximum likelihood” method of estimation, in which you choose the data that are most likely to have produced the observed value. A production run of 60 has probability 1/60 of yielding the observed serial number. A production run of 61 has 1/61 probability, a production run of 62 has a probability of 1/62, etc. A production run of anything less than 60, of course, has 0 probability of generating the observed value. Hence, you would choose 60, since it has the maximum probability of producing the (exact) observed value.