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# Laying the Odds

2 November 2003

In my last two articles I looked at the Pass Line in the game of Craps and the effect of taking single odds. This month I want to look at the Don't Pass Line and the effect of what is called laying the odds.

To begin with, let's see what is involved in the Don't Pass Line; we'll worry about odds later. A Don't Pass wager is a bet that the shooter will roll Craps on the comeout roll or will make a point and seven out. Now you might think that since this is a bet with the house that the player would now enjoy the 1.4142% edge that the casino has. Not so. The reason is that when the Craps 12 is rolled, the players don't win; rather they push. Now here is something interesting. Some writers, such as Olaf Vancura in Smart Casino Gaming, Index Publishing, 1996, quote the house edge for the Don't Pass wager as being 1.364%. Others, such as Frank Scoblete in Forever Craps, Bonus Books, 2000, state it as being approximately 1.40%. Is one right and one wrong? Nope, it's all in how you look at it. So let's look at it!

I want to postulate a player making \$60 bets on the Don't Pass Line and not laying any odds. If this player plays 1980 games (see my September article) we obtain the following tabulations.

 Event Freq. Line Bet Total Line Odds Bet Total Odds Line Pay Odds Pay Natural 440 60 26,400 0 0 -26,400 0 2 or 3 165 60 9,900 0 0 +9,900 0 12 55 60 3,300 0 0 0 0 4 Made 55 60 3,300 0 0 -3,300 0 4 Not 110 60 6,600 0 0 +6,600 0 5 Made 88 60 5,280 0 0 -5,280 0 5 Not 132 60 7,920 0 0 +7,920 0 6 Made 125 60 7,500 0 0 -7,500 0 6 Not 150 60 9,000 0 0 +9,000 0 8 Made 125 60 7,500 0 0 -7,500 0 8 Not 150 60 9,000 0 0 +9,000 0 9 Made 88 60 5,280 0 0 -5,280 0 9 Not 132 60 7,920 0 0 +7,920 0 10 Made 55 60 3,300 0 0 -3,300 0 10 Not 110 60 6,600 0 0 +6,600 0 Totals - 1980 --- 118,800 0 0 -1,620 0

Figure 1
\$60 Don't Pass with no Odds

So the player wagered \$118,800 and lost \$1,620. Thus, the house edge would be 1,620 divided by 118,800 and expressed as a percentage. The result is 1.363636…%. So is Olaf right and Frank wrong? No, not so fast. When the 12 is rolled it is a push. If the bettor opted to just leave his money on the don't Pass Line until he either wins or loses, then the 12 is a non event (such as rolling a non point after a point has been established). In this case, the total amount wagered would be only \$115,500; the loss would be the same. Dividing 1,620 by 115,500 and expressing the result as a percentage we obtain 1.4026%, Frank's number. Gaming writers have been squabbling over this for as long as I can remember. In fact, the statistician and Craps enthusiast Stuart Ethier from the University Of Utah presented a paper titled On the House Advantage, which was presented at the Fifth International conference on Gambling and Risk Taking held at Caesars Tahoe in October of 1981, in which he presented a convincing mathematical argument for throwing out ties in evaluating gambling games. On the other hand, if a Don't Pass bettor picks up his chips and leaves after pushing the 12, he is not the guy to whom the 1.4026% number is referring.

It gets worse. When it comes to laying odds (described shortly), Olaf states the Don't Pass with single odds as having a house edge of 0.682% while Frank states the figure as 0.83%. Actually, as we'll see, there are two other numbers one could report. How can that be? I'll show you, but first let's discuss how to lay the odds.

To lay the odds, one wagers against the shooter making the point and is paid fair odds. For example, if the Odds bet is \$6 and the point is 4, the bettor wins \$3 if the shooter sevens out and loses the \$6 if the shooter makes the 4. In other words, the odds are paid 1 to 2, just the opposite of the Pass Line situation we discussed last month. Similarly, 5 and 9 are 2 to 3, \$6 gets you \$4, and the 6 and 8 are 5 to 6, \$6 gets you \$5. Clearly Odds on the Don't Pass should be made in multiples of 6.

Now here is the problem. When I say I want single Odds on the point, what amount am I talking about? Well, I might mean that I want to lay the odds for an amount that is equal to my line bet. Or, I might mean that I want to make an Odds bet that will pay me an amount equal to my line bet if I win. Let's look at each of these situations.

Suppose that I first wager \$36 on the Don't Pass line and then lay \$36 odds whenever a point is made. This produces the following table

 Event Freq. Line Bet Total Line Odds Bet Total Odds Line Pay Odds Pay Natural 440 36 15,840 0 0 -15,840 0 2 or 3 165 36 5,940 0 0 +5,940 0 12 55 36 1,980 0 0 0 0 4 Made 55 36 1,980 36 1,980 -1,980 -1,980 4 Not 110 36 3,960 36 3,960 +3,960 +1,980 5 Made 88 36 3,168 36 3,168 -3,168 -3,168 5 Not 132 36 4,752 36 4,752 +4,752 +3,168 6 Made 125 36 4,500 36 4,500 -4,500 -4,500 6 Not 150 36 5,400 36 5,400 +5,400 +4,500 8 Made 125 36 4.500 36 4,500 -4,500 -4,500 8 Not 150 36 5,400 36 5,400 +5,400 +4,500 9 Made 88 36 3,168 36 3,168 -3,168 -3,168 9 Not 132 36 4,752 36 4,752 +4,752 +3,168 10 Made 55 36 1,980 36 1,980 -1,980 -1,980 10 Not 110 36 3,960 36 3,960 +3,960 +1,980 Totals - 1980 --- 71,280 --- 47,520 -972 0

Figure 2

Don't Pass Laying Odds Equal to Line Bet

If you add the numbers 71,280 and 47,520, you find that the player analyzed in Figure 2 has wagered a total of \$118,800, the same as the player from Figure 1, but he has only lost \$972. Dividing 972 by 118,800 and expressing the result as a percentage we find that the house edge has dropped to 0.818181...%. Once again, however, this calculation counted the tie on the 12 as a resolution. If one assumes that play will always continue until the game is resolved with a win or a loss, then one should subtract the 1980, listed as wagered when the 12 appeared, from the total \$118,800. The result is \$116,820. Dividing 972 by 116,820 and expressing the result as a decimal we have the house edge as 0.832%, Frank's number.

Now let us consider a different scenario. In this one the player bets \$30 on the Don't Pass and whenever a point is established lays enough odds to recover an amount equal to his \$30 line bet whenever he wins. Here is the table.

 Event Freq. Line Bet Total Line Odds Bet Total Odds Line Pay Odds Pay Natural 440 30 13,200 0 0 -13,200 0 2 or 3 165 30 4,950 0 0 +4,950 0 12 55 30 1,650 0 0 0 0 4 Made 55 30 1,650 60 3,300 -1,650 -3,300 4 Not 110 30 3,300 60 6,600 +3,300 +3,300 5 Made 88 30 2,640 45 3,960 -2,640 -3,960 5 Not 132 30 3,960 45 5,940 +3,960 +3,960 6 Made 125 30 3,750 36 4,500 -3,750 -4,500 6 Not 150 30 4,500 36 5,400 +4,500 +4,500 8 Made 125 30 3,750 36 4,500 -3,750 -4,500 8 Not 150 30 4,500 36 5,400 +4,500 +4,500 9 Made 88 30 2,640 45 3,960 -2,640 -3,960 9 Not 132 30 3,960 45 5,940 +3,960 +3,960 10 Made 55 30 1,650 60 3,300 -1,650 -3,300 10 Not 110 30 3,300 60 6,600 +3,300 +3,300 Totals - 1980 --- 59,400 --- 59,400 -810 0

Figure 3
Don't Pass Laying Odds to Produce a Win Equal to Line Bet

Adding the two totals of 59,400 we again get a total amount bet of \$118,800, just as in Figures 1 and 2. Now, however, the losses have dropped to \$810. Dividing 810 by 118,800 and expressing the result as a percentage, we obtain a house edge of 0.682%. This is the figure reported by Olaf. Once again, however, the push on 12 has been treated as a resolution. If we assume the player will always let the 12 ride until a win or loss occurs then we have to reduce the total wagered by \$1,650 to \$117,150. In this case the house edge is 0.6914%.

So we have four numbers representing the house edge for the Don't Pass while Laying Odds wager. Which is the most reasonable one? In any Craps game in which I have participated, including single odds games, I have always been able to lay enough odds to produce a win equal to my line bet. For me, that narrows it down to either 0.682% or 0.6914%. I am also firmly in the Stuart Ethier camp when it comes to ties. Counting ties as resolutions can lead to strange results. If you want proof of this refer to my article Mensa Mystery, which appeared May 4, 2003 on this web site. There I show that counting ties as resolutions leads to an absurdity. So my choice is the 0.6914% figure. This puts me in the odd duck department since I have never seen the number 0.6914% reported by anyone else.

Actually, nothing is new; I've been in the odd duck department for a long time.

See you next month.

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers