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Does Taking Down Bets Mid-Roll at Craps Lower the House's Edge?21 June 2006
The issue is especially relevant in craps, since recoverable bets often languish undecided during many hurls of the hexahedrons. Say, for instance, you Place the nine for $5. Of 36 ways dice can land, four win $7 (6-3, 5-4, 4-5, 3-6), six lose $5 (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), and the other 26 outcomes push.
By convention, edge at craps is stated based on probabilities relative to deciding
events, not the 36 ways the dice can land. So chances of winning and losing
on the nine are given as four out of 10 and six out of 10, respectively. Edge
is accordingly found as [(4/10)x$7 - (6/10)x$5]/$5, which equals 2/50 or the
familiar 4 percent. That is, the house earns an average of 4 percent of the
money at risk on the wager gone to resolution.
Assuming you leave the bet in action until it's resolved, what would edge look like per throw of the dice? The probabilities would be 4/36 of winning and 6/36 of losing. Edge would therefore be [(4/36)x$7 - (6/36)x$5]/$5 or 1.1 percent. Monetarily, in 36 ideal tosses, you'd average wins of 4x$7 and losses of 6x$5, which is $28 - $30 or -$2.00. The bet would accordingly cost you $2.00/36 or somewhat over five and a half cents per throw.
To relate 1.1 percent and 4 percent or $0.056 and $0.20, recognize that you expect bets on the nine to be decided in 10 out of every 36 throws. That's once every 3.6 tosses. The 1.1 percent times the average of 3.6 throws needed to get a decision equals 4 percent. Likewise, $0.056 times 3.6 throws yields $0.20.
Pretend you bet on the nine for $5, taking it down if it wins or isn't resolved
after six throws. Adhering to standard practice, in six as opposed to 36 throws,
you'd average (6/36)x10 or 1.67 decisions. Of these, (6/36)x4 or 0.67 would
win and earn you $7; (6/36)x6 or 1.00 would lose and cost you $5. Edge would
be [(0.67/1.67)x$7 - (1.00/1.67)x$5]/$5, or 4 percent. The theoretical loss
on your $5 at risk would be (0.67/1.67)x$7 - (1.00/1.67)x$5, which equals $2.80
- $3.00 or -$0.20. The same as for keeping the bet in action until it's resolved.
To illustrate the effect, envision two alternate strategies for betting $5 on the nine. One approach is to keep the bet in action on every throw. The second is to start after a table is cleared by a seven. On a win, the bet is taken down and not replaced until after the next seven. A loss means a seven has occurred and the bet is made again. If the bet is not decided in six throws, it's taken down and not restored until after the next seven.
A simulation was run of these strategies involving 10 million throws. Bets always in action yielded 2.8 million decisions and a net loss of $565,000, roughly 4 percent of a $14 million handle. The one-win/six-throw criterion yielded around half the action 1.4 million decisions and a net loss of $282,000, about 4 percent of a $7.1 million handle. The disparity is in quantity and not quality. In how much action players enjoyed. Of course, enjoyment is subjective. If you enjoy taking down your bets, for whatever reason, who's to argue? For, in the pronouncement of the principal poet of probability, the celebrated Sumner A Ingmark:
Where uncertainty rules,
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