'Groves replied that any general who had won five battles in a row might safely be called great. Fermi then asked how many generals are great. Groves said about three out of every hundred. Fermi conjectured that if the chance of winning one battle is 1/2 then the chance of winning five battles in a row is (1/2)5 = 1/32. “So you are right, General, about three out of every hundred. Mathematical probability, not genius.”'
(1/2)^5 = 1/32, yes, and this is very close to three out of a hundred. But is Fermi entitled to derive from this his conclusion, "Mathematical probability, not genius"? I think not. Not with any degree of certainty, anyway. The most he is entitled to say is that there is a distinct *possibility* that it is probability which is at work, and not genius.
Suppose that Groves was right, not just in his figures, but also in his assumption that there is something "great" about certain generals, which is reflected in their track record. To be specific, suppose that a "great" general will always beat one who is "non-great", regardless of any other factors such as superiority of numbers. Then suppose that we have a tournament to discover who is great and who isn't.
For a tournament the maths is easiest if the number of contestants is a power of two, so suppose that there are 128 generals, of whom four are great. 4 out of 128 is the same as 1/32, which we have already established is very close to three out of a hundred. Let's also assume that the great generals are astute enough not to face each other in battle unless absolutely necessary.
In the first round there are 64 battles, resulting in 64 victors, of which 4 are great and 60 won by chance.
The 64 defeated generals can be assumed to be removed from any further action, having been killed in battle, exiled to Elba, kicked upstairs to the House of Lords, or otherwise retired to somewhere where they can do little further damage.
In the second round there are 32 battles, resulting in 32 victors, of which 4 are great and 28 won by chance.
In the fifth round there are 4 battles, resulting in 4 victors, all of whom are great.
Note that if we have no means of discerning greatness other than track record, this whole scenario is indistinguishable from the one in which there is no such thing as greatness, only chance. The elaborate tournament, which might have been expected to decide the issue, actually tells us nothing one way or the other. The unresolved question remains unresolved.
It would have been an entirely different matter if, prior to the tournament, those judges best equipped to assess the aptitude of the untried generals had reliably pointed out the four eventual fifth-round victors, and only those four, as being destined for greatness. And it would have been yet another matter if the best judges of military aptitude had a track record of picking out the best generals in advance which was no better than chance.