[Bozeman]

This may be astronomical, but it is not astronomically small.

First, you forgot the 3/7, 2/6, and 1/5 terms at the end.

If you do the game without the special first rule, you have
48/52 .... 1/5, and all the terms cancel except
1*2*3*4
----------------
52*51*50*49

which just happens to equal C(52,4).

It is not clear to me how the rules treat when you first draw an ace, but if it means you reshuffle and try again, then the first term is 48/52*(1+4/52+(4/52)^2+(4/52)^3)...), which just happens to equal 1. If there were no aces in the deck, it would be 48/48=1.

Now the remaining terms are 24/(51*50*49*48), so the answer is 1 in 249900 (not surprisingly, C(51,4)).

Craig