A formerly affluent person came to the conclusion that people lose in roulette because they are greedy, so he made a system that is humble and anything but. He set a daily goal to win - $1,024.. and decided he'd stop playing when he comes close to that goal, to $1,000 winnings or more, or a total of 37 rounds, whichever came first. Further, he would always only aspire for half of the money he had left to win, and started out in Monte Carlo. The European roulette table has slightly better odds as there are 37 numbers instead 38; 16 are black, 16 red, and one green (the zero; American boards also have a double zero). He would bet on black each time, with 1:1 winnings and a chance to win of 18/37 which is 48.6%. Thus, on the average, he should lose.
But, he had the system to help him. On the first round he bet half of the amount yet to be won; $1,024 / 2 = $512. He lost, so now he had $1,536 to be won. On the next roll he bet $1,536 / 2 = $768, and won, so the amount to be won dropped to $768. Each time he won, the amount to be won decreased by 50%; each time he lost, it increased by 50%.
He expected to win 18 times and lose 19 times. On this day, however, he was unlucky, and only won 14 times and lost 23 times.
At the end of the day, the amount yet to be won was
$1,024 x (1 - 0.50)^14 x (1 + 0.50)^23
which is about $700. In other words, despite being on a losing streak, he ended up having won about $300, and went to the casino restaurant to mourn his bad luck by spending his winnings on martinis, steak, and lobster.
If the system wins money even on an unlucky day, what's the catch? Or is there a catch?