Originally Posted by jfc

Consider all possibilities after a pair of trials of BJ's roulette example:

-$22 = loss, loss = -10 -8
+$18 = win, win = +10 +8
+$02 = loss, win = -10 +12
+$02 = win, loss = +10 -8

Call a win, loss or loss, win pair odd. And the other same results even.

Imagine you fluked 18 consecutive odd pairs in 36 spins. You'd win $2 first, then 96% of your previous win for the next 17 pairs.

That comes to ~$130.

By that time your stake is only $5. Lose that on the final spin for a profit of $125 on 18 wins and 19 losses.

So even with more losses than wins, it is possible to win on this negative game providing you get enough odd pairs.

BJs sample conveniently has 11 odd pairs and only 7 even pairs. And that is why it ends up profitable.

An equal number of odd and even pairs is the most likely outcome, resulting in a loss, and eventual ruin for any mug persisting with such a daft staking plan.