I was happy with this until my post on this (which was almost a year ago). My ex-student, and Bayesian extraordinaire, Brendon Brewer, said something that bothered me. Namely, the chance of seeing a tank depends on the number of tanks, with more chance of seeing one if there are more of them (obviously!), and this seems to cancel out the effect of knowing the serial numbers of the tanks. So how could it work.
I had a good week and so decided to return to the problem.
Let's start you being a battlefield intelligence officer, and nearby a big battle breaks out and rages. it is too dangerous to venture onto the battle field at the moment, but reports are coming in that there is new, dangerous tank prowling the battlefield. Reports continue, and soldiers report these tanks are everywhere. Reports also come in that the soldiers are knocking out these monsters with an impressive success rate.
Eventually, the battle abates, and you venture on to the battle field, expecting to see heaps of wrecked tanks, but you find far fewer than you expected.
What's happening? If the reports were correct, there were loads of tanks and the soldiers were good and defeating them. So where are all the wrecks?
Now we can see the problem! There could have been a small number of tanks, and the soldiers could have been good at knocking them out, but equivalently there could have been a large number of tanks and the soldiers were actually pretty poor at destroying them, or somewhere in between. In all case, the number of wrecks on the battlefield would be the same.
OK, now for the technical part. While λ might be the expected number of wrecks you find, it might not be an integer number. So, how do you relate λ to the probability of seeing a number of wrecks, N, on a battlefield?
The problem was solved long ago, and what you use is the Poisson distribution. If you are not mathematical, the following may look horrible, but if, like me, you don't mind a bit of recreational mathematics, it's rather lovely.
Suppose you saw the wrecks of 27 tanks on the battlefield, what does the probability distribution of f and M look like? Now, this is a 2-dimensional problem, and so I can do it analytically, and, with the help of Matlab, you get
Adding in the next 4 serial numbers (45, 150, 213 & 58) we get this
Just to do a final check, let's assume that there is only a small number of tanks in total, 34, but that the soldiers are good at knocking them out (f=0.75). What does the probability distribution look like now?
Reverend Bayes on their side has already won the war.
Have a good long weekend!