[PairTheBoard]

I think everyone is in agreement that the Martingaler goes ahead and goes behind infinitely often with probabilty 1. I think it's also true that the martingaler spends more time being ahead than being behind. In fact, I think it can be shown that if A(n)== The number of spins the Martingaler was ahead after n spins, and B(n)== The number of spins the Martingaler was behind after n spins; Then B(n)/A(n) goes to zero as n goes to infinity with probabilty 1. However, this does not take into account the amount by which the Martingaler sometimes falls behind. I think that if we define wB(n)== The sum over all spins where he's behind of the Amounts he's behind at each spin, and wA(n) similiarly for the spins where he's ahead over n spins; Then wB(n)/wA(n) goes to -infinity as n goes to infinity with probabilty 1. Because of the House Edge.

Furthermore, if the Cosmic Casino is set up with a properly expanding pool of Martingalers, and we let T(n)== The amount the total pool of Martingalers is ahead after n spins, then P[ T(n)>0 ] goes to zero as n goes to infinity - where P[ T(n)>0 ] is the probabilty that T(n)>0.

I think these things can all be rigourously proven. I leave that to the working mathematicians here. However, even if they are proven there will still be endless debate as to what to make of it.

PairTheBoard