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# Gaming Guru

### Taking Odds in Craps

4 October 2003

Last month we analyzed the Pass Line in the game of Casino Craps. We discovered that the house edge for this bet is 1.414141…% and you saw, and I hope understood, how this wager was calculated. This month and next I want to consider a wager that is not shown on the Craps layout but is nonetheless available to players. This is the Free Odds wager. It is available to players who have wagers on the Pass Line, the Come Line, the Don't Pass Line, or the Don't Come Line whenever a point has been established. We'll talk about Don't Pass bets next month. For now I want to focus on Odds bets made for a Pass Line bet. Here is how it works.

The bet is made by placing your wager in front of the Pass Line near your Pass Line bet. Suppose, for example, that the shooter has rolled a point of 4 and you have a \$10 wager on the Pass Line. Then you can place an additional \$10 wager in front of the Pass Line near your line bet indicating that you are making an Odds bet on the 4. This is referred to as Taking the Odds. For a point of 4 you are paid 2 to 1 if the shooter makes the point; you lose the bet if the shooter sevens out. Below I show the payouts for taking the odds on various points.

 Point Odds 4 2 to 1 5 3 to 2 6 6 to 5 8 6 to 5 9 3 to 2 10 2 to 1

Figure 1
Odds for Odds Bets

In many casinos you are allowed to make Odds bets in multiples of your line bet, some as large as 100 times your line bet, and in just about all you are allowed to make a bet slightly larger than your line bet to keep the payoff from involving pennies. In the past some hardnosed casinos would pay you less than correct odds if you didn't have the proper line bet up, but with multiple odds available that situation seldom, if ever, happens. In any case, a line bet that is a multiple of \$10 works for all points and I am only going to consider single odds in this article.

Very well, let's see what sort of edge the casino has on these bets. If the point is 4 the bet can be settled by three 4s and six 7s (see my article from last month). Thus three times, on average, the player will win \$20, a \$60 win, and six times he will lose \$10, a \$60 loss. Thus, the bet is fair, that is, the casino has no edge nor does the player. The same is true of the rest of the Odds bets; all of them have an expected return of zero.

Should the player take odds? There is not quite universal agreement on this. My good friend, the late John Gwynn (see my article Remembering John Gwynn, which appeared June 3, 2001 on this web site) used to put it this way. "If you and I both play the Pass Line for the same amount of money but you take odds and I don't, then in the long run we'll both lose the same amount." John was absolutely right. Since the Odds bet has an expected return of zero, both players would end up losing the same amount, namely, 1.41414% of the money each bet on the Pass Line.

Others have a different point of view. Some argue that since a player on the Pass Line is clearly willing to make a bet with a negative expectation then that player should be willing to take a fair bet. That sounds reasonable. What is more, once a point has been established the Pass Line player is bucking the odds. For example, if the point is 9, then the probability of making the point is 2/5 whereas the probability of not making it is 3/5. Since the bet pays even money, the player is facing a 20% house advantage. By taking odds the player can reduce this edge to 10%. That sounds very reasonable. Let's look at the facts.

All references on Craps state that the house edge with single odds is 0.8485%. At the end of this article I'll show you a quick way to calculate that number, but for now I want to do things by the long, tabular method.

To keep everything in integers I want to postulate two different scenarios. One is a player making a straight \$50 Pass Line bet and not taking odds. The other is a player making \$30 line bets and taking odds when they are available. Assuming 1980 games (see last month's article) the first player generates a table that looks like this:

 Event Freq. Line Bet Total Line Odds Bet Total Odds Line Pay Odds Pay Natural 440 50 22,000 0 0 +22,000 0 Craps 220 50 11,000 0 0 -11,000 0 4 Made 55 50 2,750 0 0 +2,750 0 4 Not 110 50 5,500 0 0 -5,500 0 5 Made 88 50 4,400 0 0 +4,400 0 5 Not 132 50 6,600 0 0 -6,600 0 6 Made 125 50 6,250 0 0 +6,250 0 6 Not 150 50 7,500 0 0 -7,500 0 8 Made 125 50 6,250 0 0 +6,250 0 8 Not 150 50 7,500 0 0 -7,500 0 9 Made 88 50 4,400 0 0 +4,400 0 9 Not 132 50 6,600 0 0 -6,600 0 10 Made 55 50 2,750 0 0 +2,750 0 10 Not 110 50 5,500 0 0 -5,500 0 Totals - 1980 --- 99,000 0 0 -1,400 0

Figure 2
\$50 Line Bet - No Odds

As you can see a total of \$99,000 was wagered and \$1,400 was lost. The house edge is:

House Edge = 1400/99000 = 1.414141...%

Now let us construct the same table for a \$30 player who takes odds whenever they are available:

 Event Freq. Line Bet Total Line Odds Bet Total Odds Line Pay Odds Pay Natural 440 30 13,200 0 0 +13,200 0 Craps 220 30 6,600 0 0 -6,600 0 4 Made 55 30 1,650 30 1,650 +1,650 +3,300 4 Not 110 30 3,300 30 3,300 -3,300 -3,300 5 Made 88 30 2,640 30 2,640 +2,640 +3,960 5 Not 132 30 3,960 30 3,960 -3,960 -3,960 6 Made 125 30 3,750 30 3,750 +3,750 +4,500 6 Not 150 30 4,500 30 4,500 -4,500 -4,500 8 Made 125 30 3,750 30 3,750 +3,750 +4,500 8 Not 150 30 4,500 30 4,500 -4,500 -4,500 9 Made 88 30 2,640 30 2,640 +2,640 +3,960 9 Not 132 30 3,960 30 3,960 -3,960 -3,960 10 Made 55 30 1,650 30 1,650 +1,650 +3,300 10 Not 110 30 3,300 30 3,300 -3,300 -3,300 Totals - 1980 --- 59,400 --- 39,600 -840 0

Figure 3
\$30 Line Bet with Odds

If you add the \$59,400 in line bets and the \$39,600 in odds bets you find that the total wagered in these 1980 games is \$99,000, the exact same figure as for the player in Figure 2. Now, however, the loss is only \$840. The house edge is:

House Edge = 840/99000 = 0.848484...%

So there you have it. If two players make the same line bets and one of them makes additional odds bets they will lose exactly the same amount in the long run. However, if one of the two players makes line bets and takes odds in such a fashion that his total wagers are the same as the other player, then he will lose less in the long run.

I mentioned earlier that I would show you how to arrive at the 0.8485 figure without constructing these large tables. The Pass Line has a 1.41414...% house edge per game. Adding Odds does not change this per game edge since, as is clear from the last column in Figure 3, it is a zero expectation wager. What does change is the average bet per game. One third of the time the wager is 1 unit and 2/3 of the time the wager is 2 units. The average bet is, therefore

Average Bet =1 x 1/3 + 2 x 2/3 = 5/3

If you divide 1.414141... by 5/3, same as multiplying by 3/5, you get exactly 0.84848484...%. The house edge for multiple odds can be calculated in the same way. For example, Pass with double odds has an average bet of 7/3 and a resulting house edge of 0.60606...%.

Next month we'll look at the Don't Pass line and see the effect of laying the odds. There is lots of disagreement related to this topic; I think you'll find it amusing. See you then.

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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