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What bets offer the most chance to reach win goals before hitting loss limits?

23 January 2012

Some solid citizens visit casinos intending to risk modest bankrolls in shots at a day’s pay. They figure their prospects are decent if they set conservative profit targets and quit as soon as they reach or surpass them. This notion is generally correct. But it’s not the whole story.

Ideally, were bets to have no edge, the probability of hitting a target before biting the dust would precisely equal the amount with which a player starts – the bankroll, divided by how much the person will have by achieving the win goal – the bankroll plus the profit. So, for instance, to double a \$100 stake – earning \$100 to end with \$200, the chance of heading home happy is 100/200, which is 0.50 or 50 percent. With a \$50 win goal on the same stake, the likelihood of prevailing is 100/150, which is 0.6667 or 66.67 percent. Moving in the other direction, setting a \$200 profit target, the outlook becomes 100/300, which is 0.3333 or 33.33 percent.

The probability of attainment being inverse to the ratio of starting to desired ending levels holds in the real world where the casino has an edge. The effect of this statistical advantage is to lower players’ prospects at any level of initial and final amounts. The reduction isn’t straightforward, though. It depends on the parameters of the bets being made. These are bet size, payoff ratio, and chance of winning. Together, the latter both determine not only the edge but also the volatility.

To picture the effect, assume you play double-zero roulette and are trying to decide whether to bet on a four-number corner with a 4/38 (10.53 percent) chance to win 8-to-1, a six-number double-street with a 6/38 (15.79 percent) chance to win 5-to-1, or a 12-number column with a 12/38 (31.58 percent) chance to win 2-to-1. The payoffs and the probabilities of winning shift in tandem, such that the edge is the same in all cases, 5.26 percent.

Say you have a \$100 bankroll, are comfortable betting \$5 per spin, and want to earn \$100. Ideally, your chance of triumph is 50 percent. However, owing to the edge and other parameters, the actual probabilities are 43 percent on the corner, 39 percent on the double street, and 25 percent on the dozen. Were you to set a \$50 profit target, your outlook would be 62 percent on the corner, 59 percent on the double street, and 48 percent on the column. With a more ambitious \$200 earnings objective, your chances become 25 percent on the corner, 20 percent on the double street, and 8 percent on the column. Based strictly on the win goal versus loss limit criterion, the higher-paying bets with the lower hit rates would be the more appropriate choice.

Were you able to calm the butterflies in your stomach and bet \$10 with the same bankroll and the aim of doubling your money before busting out, your probabilities of success would be 47 percent on corners, 45 percent on double streets, and 37 percent on dozens. Your prospects would likewise improve for the smaller and larger win goals. Raising your bets enhances your outlook when reaching a win goal before hitting a loss limit is your sole criterion.

What happens if the edge varies among the alternatives? At craps, for example, you may want to compare Place bets on alternate numbers. Payoffs and expected frequencies of wins on these bets differ in a manner that influences edge as well. A \$5 four has a 3/9 (33.33 percent) chance to win \$9 and 6.67 percent edge. A \$5 five has a 4/10 (40.00 percent) chance to win \$7 and 4.00 percent edge. And a \$6 six (the minimum booked on this number at a nominal \$5 table) has a 5/11 (45.45 percent) chance to win \$7 and 1.51 percent edge.

Pretend, as with the roulette illustrations, you have a \$100 bankroll and are hell-bent on going for broke in an attempt to double your money Placing the four for \$5, you’ll come through in 18 percent of your sessions. With \$5 on the five, your chance would be 24 percent. And venturing \$6 on the six, you’d be up to 39 percent. Here, contrary to the trend at roulette, the highest-paying lowest-probability option – the four – affords the dimmest hope of joy, while the converse is true for the lowest-paying highest-probability bet – on the six; the bet on the five is intermediate. The flip-flop relative to the roulette situation occurs because all the bets cited for that game have the same edge, while in craps, the house advantage on the four is greater than that on the five, which in turn exceeds that on the six. Chances would improve across the board were you to lower your win goal and deteriorate if you raise it, but the trend would be the same. Similarly, higher bets would increase your likelihood of success but wouldn’t alter the order of rankings associated with the alternate propositions.

Wouldn’t it be wonderful if one hard-and-fast rule covered all situations? Wouldn’t the footwear industry be thrilled if one size shoe fit all feet? Then neither punters nor podiatrists would be inclined to extrapolate from this advice from the beloved bard, Sumner A Ingmark:

Gamblers who are inexorable,
Leave casinos feeling horrible.