The problem is quite simple. The enemy are producing tanks, and each has a sequential serial number (for simplicity, let's assume that the numbers are reset every month). You encounter this scene on the battle field;
This is the problem that faced the Allies in WWII; you really wanted to know how many panzers are out there. Intelligence officers were reporting productions of more than a 1000 tanks per month, but based on statistics, the predicted number was significantly fewer than that, in the hundreds. After the war, the numbers were checked against records and the statistical answer was amazingly correct (read the wikipedia page for more details).
But let's see the how we can calculate this. We'll adopt the Bayesian approach (because that's the correct thing to do it :). So, let's assume that the number of tanks actually made is a number N, and let's assume that we guess that the maximum number of tanks that could be possibly be made is M (we'll insert some real number in here soon).
On the battle field, we find a tank with a serial number, A. What is your estimate of the number of tanks made in that months (let's call this X)? We want to make a probability distribution, and where this peaks, this is our best estimate for the number of tanks build.
Clearly, the minimum number of tanks is A (because you have the serial number you have). What about the rest of the probability distribution? If you think about it, if the total number of tanks is X, then the probability of randomly selecting tank A is simply 1/X. So the probability distribution look like this
Now for the cool part. You hear a report that another tank has been knocked out, this time serial number 91. You might think that tells you nothing new, as you know the minimum number is 217, but 91 has a similar probability distribution to 217, and to get the resultant distribution for the total number of tanks you multiply these together.
I've brushed over some of the key Bayesian words and concepts here, but this is basically what it boils down to; we get more evidence and we update our beliefs. So, what's the result of now finding tank 91? The result is the red curve below.
Reports come in that three more tanks have been knocked out, 256, 248 and 61. What's the resultant distribution look like?
Report come in of 5 more tanks, number 250, 172, 189, 29 and 170. What's the distribution now?
We can continue to play this game, and with 25 tanks knocked out, we get
Now I think that is cool. And that's how information should be used.