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Best of Donald Catlin

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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.

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