|
If you had an infinite bankroll and there were no limits on how much you can bet, the strategy of betting X, then 2X, then 4X, etc., will win you X * payout when you win (which will happen eventually), as long as time is finite.
For example, bet 1 on black, paying 1 to 1. Lose. Bet 2. Lose. Bet 4. Lose. Bet 8. Lose. Bet 16. Lose. Bet 32. Win 32. Total losses = 1+2+4+8+16 = 31. Total winnings = 32. Net profit = 1.
Let's assume all 100,000 customers are playing this system, and they are bankrolled to double until $512, or 10 bets. The chance of someone losing 10 bets on a color in roulette in a row is .525 ^ 10, or .00159. This means that 159 of our wily customers per day will each lose 1023 dollars for a net to the casino of $162,657, more than covering the $99,841 the house pays out to its reasonably lucky patrons.
However, if each customer could afford 20 bets, the house would lose most days. This is because the chance of losing 20 bets in a row on a color is .00000253, meaning 253 people out of 100,000,000. However, during every 10 days the house is open, around 2.53 customers will lose all 20 bets, costing each of them $2,097,151, for an EV for the house/10 days of $4,305,794.56 ($5,305,792.03 - $999,997.47). This is an even bigger win for the house, but notice that now they have to wait longer for it. Also, for some samples of 10 days, they will not recoup their losses because not one customer will lose all 20 bets.
If each customer could afford to double 1000 times, and there were no limits on the tables, the chance of someone losing is 1.44312319 * 10^-280. So, even if someone played this game every second from the dawn of the universe to the projected date of the big bang, around 3.2 * 10^21 seconds, the chances are almost nil that the casino would recoup one dollar on the game (although with infinite time, the edge is still there). Of course, if every roulette player was carrying around a bankroll of 2.143017 * 10^301 dollars, or 10^173 times the number of neutrons that could be jammed in the universe, I guess the laws of time and space are right out the window.
In conclusion, if casinos limit their top bet on one color of roulette at anywhere less than 134,217,728 or so, even if everyone in the world could afford to drop 268,435,455 on the table, 2.8 in a hundred million (not unreasonable for the life of a big casino) would more than foot the bill for the rest of the winners.
|