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# What Are Your Chances of Winning \$100?

2 December 2003

You're off to the casino with a modicum of mad money in your fanny pack and what seems a sensible goal in your head. You want to win some amount, say \$100. Your non-sporting friends contend that nobody can beat the bosses. You know better. You know it's a matter of chance. Only, you're not sure what the chance is. And you also believe you can influence that chance by the way you gamble, although you're a tad hazy about exactly how to do it.

Were there games with no edge for either casino or player, the probability of reaching some profit point before depleting a given stake would depend on the win goal and loss limit alone. It would equal your starting bankroll divided by the total with which you want to finish. Take that \$100 goal. On a \$50 ante, this chance is 50/(50+100) or 33.3 percent. Come with a \$100 stake and it's 100/(100+100) or 50 percent. Begin with \$400 and the likelihood you'll earn \$100 is 400/(400+100) or 80 percent.

Even if the house has an edge in your game, this elementary formula is still instructive because it tells where your prospects are capped. Your chance of achievement can approach the ideal, but can't exceed it. Therefore, inquiring minds will want to know how to get as near as possible, and how near that is.

Once edge does rear its ugly head, it becomes a factor. As do average bet size and volatility. The precise relationship is more complex than back-of-napkin arithmetic. But, as a rule of thumb, you get closer to the best theoretical chance as house advantage decreases, volatility rises, and bet size goes up. Depending on the game, you can control one or another -- or a combination -- of these elements to tune session promise to your priorities.

Here's an example. Pretend you have \$200 and will play until you win \$100 or go bust. With no edge, your chance of victory tops off at 200/(100+200) or 67 percent. Now, account for reality.

Suppose you play blackjack with perfect basic strategy. Edge is 0.5 percent; "standard deviation," the way statisticians measure volatility (think of it as the average change in your fortune per round per dollar bet), is 1.12. If your wager is always \$5, the likelihood you'll quit happy is more than 61 percent. At \$10, chances improve to almost 64 percent. Vary your bet -- \$5 half the time, \$10 and \$25 a quarter of the time each. This puts the average at \$11.25 and raises standard deviation to 1.38. Betting \$11.25 flat, chances of joy would be over 64 percent; the higher volatility induced by the progression brings it above 65 percent.

Contrast these figures with what faces a fumbler who flouts "the book" and gives the house 1 percent edge. All else being the same, this person's probabilities of winning \$100 on a \$200 bankroll are below 56 percent betting \$5, roughly 61 percent at \$10, and pushing 64 percent with the \$5/\$10/\$25 progression.

Or, take roulette. The "American" version has 2.6 percent edge on even-money outside bets when 0 or 00 lose only half, and 5.26 percent on all other wagers but one. Standard deviations range from 0.98 outside, up to 5.76 on a single spot paying 35-to-1. The chances of earning \$100 with a \$200 bankroll, betting \$5 per spin, are 31 and 64 percent, outside and one spot, respectively. At \$10, the probabilities would rise to 48 and 66 percent. Notice how volatility trumps edge. Wagering on a double row (six numbers paying 5-to-1) is between the outside and single-spot extremes. Edge is still 5.26 percent but standard deviation is 2.19. Flat betting \$5 and \$10 on every round, the chances of reaching the \$100 goal on a \$200 bankroll are 51 and 59 percent, respectively.

So, you improve your chances of reaching a win goal before going belly-up by lowering edge while raising bet size and volatility. However, many solid citizens prefer to maximize their playing time, and target a session duration rather than an earnings level. This objective suggests a different strategy. Low edge is still desirable, but smaller bets and less volatility are in order. Balancing win goal and playing time is a classic enigma, artfully articulated by the punters' poet, Sumner A Ingmark:

Were e'er there simpler times and places,
When easy answers fit all cases?