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Best of Alan Krigman
Do your gambling prospects improve if you bet more often?8 February 2010
The longer a game you play, the greater your chances of a successful round. But this isn't necessarily equivalent to the likelihood of earning a profit. To see the effect, pretend you flip a coin, heads wins and tails loses. The chance of winning any individual toss is 50 percent.
You can approach a multi-round game intending to play until you win or bust out. If you bet $1 per flip and are willing to go up to five times, chances and impacts on bankroll are as follows.
• Win round 1: 50.000% chance of winning $1. • Lose round 1, win round 2: 25.000% chance of breaking even. • Lose rounds 1 and 2, win round 3: 12.500% chance of losing $1. • Lose rounds 1 through 3, win round 4: 6.250% chance of losing $2. • Lose rounds 1 through 4, win round 5: 3.125% chance of losing $3. • Lose all five rounds: 3.125% chance of losing $5.
Overall, the probability of obtaining heads totals 96.875 percent. But only a win on the first flip yields a profit. The remainder represents a break-even – 25 percent on the second try, or a loss -- 25 percent on the third through fifth flips.
"Martingale" betting supposedly overcomes the nuisance of winning a round but finishing behind. With this system in an even-money game, you double your wager after every loss. You earn one unit, in this case $1, when you finally win. And your chances are good. The probability of losing five in a row is only 3.125 percent.
Martingale betting breaks down because doubling wagers results in a "geometric progression" in which the required amounts escalate rapidly. Starting at $1, the fifth round calls for $16. A loss puts you out $31 trying to win $1. The probabilities are 96.875 percent you'll win the $1 hitting within five flips and the complementary 3.125 percent you'll miss all five and lose $31. Going more rounds pushes the effect to greater extremes. Many solid citizens would put up $1 on a 0.1 percent shot at winning $1,023. But few folks would risk $1,023 for a 99.9 percent chance of winning $1. These are the 10th-round Martingale figures.
Another alternative for a five-flip game is to play through, $1 per try, regardless of what happens along the way. The total bet is $5. Possible outcomes, and their probabilities are as follows.
• Win none, lose five: 3.125 percent chance of losing $5. • Win one, lose four: 15.625 percent chance of losing $3. • Win two, lose three: 31.250 percent chance of losing $1. • Win three, lose two: 31.250 percent chance of winning $1. • Win four, lose one: 15.625 percent chance of winning $3. • Win five, lose none: 3.125 percent chance of winning $5.
Flipping coins for even money has no edge for either side. In real casino gambling, the house has an advantage. To see how this changes the prospects of a game in which you bet in multiple rounds, consider wagers on Red at single-zero roulette. The chance of a win on any coup is 18 out of 37, or 48.65 percent; that of a loss is 19 out of 37, or 51.35 percent.
The amounts you can win or lose on Red, adhering to any of the indicated strategies, are the same as they would be flipping coins. The differences are in the chances of the various results.
For instance, five-round Martingale betting on Red, starting with $1, has 96.429 percent chance of winning $1 versus 3.571 percent chance of losing $31. Compare these figures to 96.875 and 3.125 percent in the zero-edge game, respectively. Playing five spins through, chances range from 3.571 percent for losing $5 to 2.725 percent for winning $5; the analogous probabilities for the coin toss are 3.125 percent either way.
You may not think these small percentage differences are worth worrying about. In fact, they're the engines that drive profits into the casinos' coffers. The key is that most gamblers, as individuals, barely notice the effects but they represent big bucks when applied to the money wagered over time by high numbers of players. As the poet, Sumner A Ingmark, penned:
So folks can win despite the tilt.
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