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Best of Donald Catlin  # The Horn Bet

5 July 2008

A couple of weeks ago one of my colleagues asked me if I knew why the Horn bet in craps has such strange odds. I had to confess that since I never bet the Horn I didn't even know what the odds were. "17 to 4" he told me. "You're right," I said, "those are peculiar odds and yours is a reasonable question; I'll look into it."

Let me give you a bit of background. The Horn bet is a wager that 2, 3, 11, or 12 will occur on the next roll of the dice. For obvious reasons this is also called the Craps - Eleven bet. These numbers can also be bet on separately. The 2 and 12 each pay 30 to 1 and the 3 and 11 each pay 15 to 1. Let's see what happens if we put one unit on each of these four numbers.

The table below indicates how we can calculate the expected return on the aforementioned wager. Notice that if we roll a 2, for example, we receive 30 units but we lose three units on the 3, 11, and 12. Our net gain would be 27 units. Similarly, if we rolled a 3 our net gain would be 12 units. Here is our table:

 Event Prob. Payoff Product 2 1/36 27 27/36 3 2/36 12 24/36 11 2/36 12 24/36 12 1/36 27 27/36 Lose 30/36 -4 -120/36 Total 1 -- -18/36

Note that 18/36 is the same as 1/2 so our expected return is -1/2.

There are 6 out of 36 ways to win the Horn wager, so the probability of winning is 1/6. The probability of losing is, therefore, 5/6. Suppose that we place 4 units on the Horn and set the payoff so that the expected return is -1/2 as it was above. How do we do this? We just let x represent the payoff on a four unit wager and construct the following table.

 Event Prob. Payoff Product Horn 1/6 x x/6 No Horn 5/6 -4 -20/6 Total 1 -- (x-20)/6

So we simply set (x-20)/6 = -1/2 and solve for x. Easy! Multiply through by 6 and we have x - 20 = -3. Add 20 to both sides and obtain x = 17. There's you answer. If we put the expected return on a per unit basis we have to divide -1/2 by 4 and obtain - 1/8, which is the same as -12.5%. If you check any standard reference on craps (for instance Frank Scoblete's book Forever Craps) you'll see the house edge for the Horn bet listed as 12.5%.

I hope this clears up the seemingly strange 17 to 4 odds on the Horn bet. See you next month.

Don Catlin can be reached at 711cat@comcast.net

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Best of Donald Catlin
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