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Here's a Simple Rule to Estimate your Chances of Success1 September 2002
True, you may not define a loss limit and win goal explicitly. And, some individuals keep augmenting their stakes trying for a recovery at the low end, or play as long as they can while hoping their luck continues at the high. Still, perceiving intuitively how chance is related to profit and loss levels can help you establish a rational gambling style that suits your personality.
To see how this works, consider the most basic of gambles - the flip of a coin. If you understand this simple paradigm, you know everything important about casinos. The rest is window dressing.
Make believe 100 solid citizens enter a coin flipping tournament. They oppose one another, each with a dollar. The game runs until everyone either wins or loses a dollar. Half will succeed, half fail. There's no other way to meet the exit criterion. The chance of multiplying the stake by two and ending with $2, rather than going bust, is therefore 50 winners out of 100 starters. That's 50/100, which equals one out of two or 50 percent.
Same situation, except that everyone plays to win $3 or lose $1. The only possible finale is that 25 players succeed and 75 fail. That is, 25 winners at $3 each get a total of $75, with the money coming from 75 losers at $1 each. No other allocation works. The probability of success, of multiplying a bankroll by four and ending with $4, as opposed to losing $1, is 25 out of 100. That's 25/100, which equals one out of four or 25 percent.
Again, hold the game but change the objective to win $99. Now, one person will triumph and 99 will tank. The probability of multiplying a stake by 100 is 1/100, one out of 100, 1 percent.
Go in the other direction. Start with $1 but bet $0.01, $0.05, or $0.25 per flip and terminate with a quarter profit or a dollar loss. Now 80 winners will score $0.25 each, a total of $20 in prize money. This will come from 20 losers, each contributing $1. The chance of finishing with $1.25, multiplying a bankroll by 5/4, is 80/100. That's 4/5 - an 80 percent success rate.
You can see the pattern. Call it the "rule of inverses." With no house edge, the chance of reaching a win goal before going broke is the inverse of the factor by which you want to multiply your bankroll. The odds lengthen as you try for more. They shorten as you go for less - shifting in your favor when you shoot for a profit smaller than the downswing you're willing to endure.
Start with $100. In an "ideal" no-edge game, the probability of earning $900 and reaching $1,000 - multiplying a stake by 10, would be 1/10 or 10 percent. The chance of making $4,900 and reaching $5,000 - multiplying a $100 bankroll by 50, is 1/50 or 2 percent. The likelihood of earning $40 and reaching $140, multiplying a buy-in by 1.4, is 1/1.4 or just over 71 percent.
How useful are these figures? This depends on the specifics of the game and the size of your bets. But for practical cases, approximations based on coin flips are at least in the ballpark, and they clearly put a ceiling on your chances of success.
Pretend you play blackjack with perfect Basic Strategy at $10 per round, starting with $200. You want to double your money - win $200 and quit with $400. You have 46 percent chance of success, compared to 50 percent predicted by rule of inverses. Betting $5, your chance drops to 42 percent. Still, not all that far from the 50 percent "best case" estimate. Confirming what that bard beloved of artful bettors, Sumner A Ingmark, said about the educational value of simplifying assumptions:
There's much to learn by flipping coins,
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