Sometimes Smart People Make Dumb Mistakes

A reader e-mailed me to point out that the book discusses a combination bet to be used on tables that pay 3-1 on the field when a 12 is rolled, instead of the 2-1 return on 12 the field pays at some casinos. Make a $5 place bet on 5, $6 place bets on 6 and 8 and a $5 wager on the field, and the Mensa Guide says the house edge is 1.136 percent.

That's a figure that would shock game analysts, since all the individual pieces of this combination have higher house edges than 1.136 percent. House edges are 4 percent on the place bet on 5, 1.52 percent on the place bets on 6 and 8 and 2.78 percent on the field when rolling a 12 brings a 3-1 payoff.

The reader noted that in my Craps Answer Book, I'd written that the house edge on a combination was a weighted average of all bets in the system, and couldn't be lower than the edge on the best individual bet. So how could this system defy the math?

The reasoning that would lead to the 1.136 percent figure goes like this. See if you can find the flaw.

Imagine a perfect sequence of 36 rolls of two dice in which each possible combination comes up once. On each roll we risk $22 -- $5 on the 5, $6 each on 6 and 8 and $5 on the field. So our total risk is $792.

Here are our returns:

• On the one time we roll 2, we win a 2-1 payoff on the field, bringing $10 in winnings. We lose no wagers. Profit: $10.
• On the two times we roll 3, we win even money on the field. We lose no wagers: Profit: $10.
• On the three times we roll 4, we win even money on the field. We lose no wagers. Profit: $15.
• On the four times we roll 5, we win 7-5 payoffs on our place bet on 5, for $28 in profits, but lose our field bets. Profit: $8.
• On the five times we roll 6, we win 7-6 payoffs on our place bet, for $35 in profits, but lose our field bets. Profit: $10.
• On the six times we roll 7, we lose everything. Loss: $132.
• On the five times we roll 8, we win 7-6 payoffs on our place bet, for $35 in profits, but lose our field bets. Profit: $10.
• On the four times we roll 9, we win even money on our field bet. We lose no wagers. Profit: $20.
• On the three times we roll 10, we win even money on the field. We lose no wagers. Profit: $15.
• On the two times we roll 11, we win even money on the field. We lose no wagers. Profit: $10.
• On the one time we roll 12, we win a 3-1 payoff on the field. We lose no wagers. Profit: $15.

Add up all the profits, and you get $123. We lose $132 on the 7s, so our overall loss comes to $9. Divide $9 in losses by $792 in wagers, and you get .01136. Multiply by 100 to convert to percent, and the house edge for the combination in 1.136 percent.

Or so it seems.

Those of you who have followed along as I've calculated craps edges before may have noticed something. This calculation assumes that we bet a fresh $22 on every roll of the dice. But that's not what we do when we make place bets. Most of the time, our place bet neither wins nor loses. When we place the 6, we win if a 6 rolls and lose if a 7 rolls, but on any other number, it just stays active unless we choose to take it down.

Essentially choosing to take the place bets down on every roll instead of letting them play out to a decision grossly overstates the amount of money put at risk. In fact, the real risk is $362 --- less than half the $792 used above, because it take a bit more than two rolls to settle the average place bet.

If we used that method to calculate the house edge on a place bet on 6, taking down the bet and starting fresh on every roll, we'd come up with an artificially low edge of 0.46 percent, instead of the 1.52 percent used by analysts everywhere, including the Mensa Guide.

So it goes with this combination. The listed house edge is artificially low. Calculated correctly, it comes to 2.5 percent --- higher than the house edges on the 6 and 8, but lower than the house edges on 5 and the field. It's a weighted average, just as it has to be.

Nothing against the Mensa Guide, which overall is pretty good. But suggesting that hedges and combinations can lower the house edge -- well, that's just not smart.