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Best of Alan Krigman

Gaming Guru

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How to Use Expected Value to Weigh Craps Options

12 November 2001

There's probably no scientific study of the matter. But the lion's share of the money wagered at craps is likely on Place bets. Certainly, after the come-out. For the uninitiated, these are bets on four, five, six, eight, nine, or 10. They win if the number hits and lose if the seven rears its ugly head.

The odds players fight and the amounts they're paid for winning differ among numbers. On four or 10, odds are 2-to-1 and payoffs are 9-to-5. On five or nine, odds are 3-to-2 and payoffs are 7?to-5. On six or eight, odds are 6-to-5 and payoffs are 7-to-6.

Numerology notwithstanding, how do solid citizens select which boxes to Place? Some folks like the superior payouts on "outside" numbers - often, but not always, knowing that greater returns mean overcoming steeper odds. Others prefer the higher likelihood of wins on the "inside," accepting the correspondingly lower earnings. A few mind their math and try to minimize the house's edge - opting for 1.5 percent on six and eight as opposed to 4.0 percent on five and nine or 6.7 percent on four and 10.

Statistically, bets which give the casinos the least edge are the most favorable for players. But the impact of edge is seen through a glass only darkly. It's easy to picture a $7 payoff for a $5 bet. Tougher, but not extremely so, to discern that sixes and eights tend to win more frequently than fours and 10s. But only a Nikolai Ivanovitch Lobatchevskii could belly up to a craps table and think, much less care, about 6.7, 4.0, or 1.5 percent.

"Expected value" conveys the same information as edge, but stated in dollars and cents rather than percentages. Essentially, expected value is what a bet is worth prior to its resolution, factoring in chances and amounts of winning and losing.

Expected value is simple to figure for Place bets. Multiply the probability you'll win times the amount you'll have - the bet you get back plus the payoff - if you do. Here's how it works:
* Four or 10: Three ways to win and six to lose, so probability of winning is 3/(3+6). Bet $5 to win $9 and have a total of $14. Expected value of $5 is therefore (3/9)x($14)=$4.67.
* Five or nine: Four ways to win and six to lose so probability of winning is 4/(4+6). Bet $5 to win $7 and have a total of $12. Expected value of $5 is therefore (4/10)x($12)=$4.80.
* Six or eight: five ways to win and six to lose so probability of winning is 5/(5+6). Bet $6 to win $7 and have a total of $13. Expected value of $6 is therefore (5/11)x($13)=$5.91.

One way to use this information would be to reduce each answer to expected value per dollar bet, then compare options. Per dollar bet, you'd get $4.67/5 or just over $0.93 for four or 10, $4.80/5 or $0.96 for five or nine, and $5.91/6 or about half way between $0.98 and $0.99 for six or eight. Further, subtracting these values from $1.00 shows the theoretical cost of making the bet. Again, per dollar bet, it's almost $0.07 on four and 10, $0.04 on five or nine, and between $0.01 and $0.02 on six and eight.

It may be even more helpful to find the difference between the amount actually at risk and its expected value, then ask whether you believe it's worth this theoretical price to take a shot. Are you willing to spend $5.00 - $4.67 = $0.33 to try to collect $9 on a $5 four or 10? Is it OK to pay $5.00 - $4.80 = $0.20 to go for $7 on a $5 five or nine? How do you feel about $6.00 - $5.91 = $0.09 to shoot for $7 on a $6 six or eight?

What's right for you? Gaggles of gambling gurus grimace at giving the house more edge than need be to get into the game. But people gamble for different reasons, and who's to argue with motivation? Realistically, what's best for each person represents a balance - the gain sought versus the cost to try. The polyrythmic poet, Sumner A Ingmark, agreed, adding this appropriate apperception:

If everybody's doing it,
There's not much sense eschewing it,
So, rather than just ruing it,
Explore the sense imbuing it.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.