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Best of Alan Krigman
Some solid citizens study games by comparing different wagers to be sure of putting their dough on the layout in the most propitious way. This is certainly a good idea... provided they understand the yardsticks they're using to make their judgements.
Comparison across games is often a matter of apples and oranges. It's not always straightforward even within a game, either. In craps, for instance, consider the nominally straightforward alternatives of Place bets on the four, five, and six.
Most experts rank bets by the house edge. The usual figures are 6.7, 4.0, and 1.5 percent of the amounts bet on the four, five, and six, respectively. These quantities are based only on rolls of the dice when wins or losses occur. However, of the 36 ways dice can land, only nine yield decisions on the four (three fours and six sevens), 10 on the five (four fives and six sevens), and 11 on the six (five sixes and six sevens).
A more reliable comparison might therefore be made by referencing edge to all rolls of the dice, not just those when the number or a seven occurs. Per-throw, edge is 1.67, 1.11, and 0.46 percent for the four, five, and six, respectively.
The waters are muddied because Place bets on the six are in multiples of $6, whereas the other numbers take multiples of $5. Gamblers worry about losing dollars, not percentage points. To find the monetary equivalents of edge, multiply the percentages by the amounts bet. For minimum wagers at a $5 table, results are $0.33 on the four, $0.20 on the five, and $0.09 on the six on a per-decision basis; the corresponding per-roll values are $0.083, $0.055, and $0.027. Theoretical loss due to edge for the same bets might also be stated for sessions where results were "statistically correct." In 288 throws of the dice, roughly three hours, the sums are $24 on four, $16 on five, and $8 on six.
What about the 25, 26, and 27 throws out of 36 that yield no action? The bosses want lots of settlements since, in theory, they make money on every one. Conversely, boosting the number of bets decided during a session should be deleterious to players because it tends to drain more money from their bankrolls.
Some gamblers want as much action as possible. Others prefer to simply log the time they believe is the key to earning comps at the tables. Who's to say these attitudes are right or wrong?
Additional factors making these bets difficult to compare are the differences in the amounts vulnerable to loss and in the payoffs for wins. The six has $6 up for grabs versus $5 on the five and four. Likewise, the six and five pay $7 versus $9 for the four.
The chance that a given bankroll will suffice for some number of throws is metric that accounts for the volatility caused by the sizes and frequencies of wins and losses as well as the edge. Say the stake is $100 and bets are working for each of 288 throws. For a $6 six, $100 would be enough 88 percent of the time. With a $5 five, the probability rises to 90 percent. A $5 four cuts the chance of remaining in action to 85 percent. Five would be best, followed by six then four, although the distinctions are small.
The chance of reaching a win goal before a loss limit is another criterion covering both edge and volatility. Pretend you want to double $100 or go bust trying. With the $5 and $6 bets, you've a 40, 24, and 18 percent shot on six, five, and four, respectively.
Which of these bets is best? Most of the quantitative ratings suggest that the six would be the best choice, followed by five and then four. Still, solid citizens aplenty play their hunches, prefer longshots with high payoff ratios regardless of the low probability, disbelieve that math has anything to do with whether or not they'll get lucky, or have a system requiring them to take a certain position. Will you tell them that the four is a sucker bet? It'd be wiser to remember that any casino wager can win or lose. After all, someone hits a megabucks jackpot now or then. Moreover, as the beloved bard, Sumner A Ingmark, reminded us:
We lack a single scale to measure,