Knuth proved this path product estimate is quite good in the following technical sense.
Let G be the set of all possible short chess games. We want to know |G| , the size of G . The issue is |G| is so large that we can not hope to directly enumerate all of the elements of G to directly perform the counting.
Theorem: (1 / |G|) ∑ g in G p(g) = |G| .
This is traditionally written in terms of the expected value operator as E g in G [p(g)] = |G| .
Knuth considered this surprising enough that he wrote: “We shall consider two proofs, at least one of which should be convincing.”