[PairTheBoard]

btw, If you insist on the proportional betting being strictly less than 100% of the Bankroll, then in the above example make it 99%. The expected value of your bankroll at time t=n flips is then greater than the RHS below:

E[Bn] > .55^n * 1.99^n = 1.0945^n

Now you never go broke, the "Growth Rate" of your bankroll is negative, your bankroll still converges in probabilty asymptotically to zero, but the Expected Value of your Bankroll grows exponentially toward infinity.

PairTheBoard