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# Odds

8 December 2007

I was recently working a crossword puzzle that contained the word "probability" as the clue for a four letter word. The answer turned out to be "odds." I'm used to crossword creators providing questionable clues and this example certainly falls into that category. Well almost. Strictly speaking odds are not about probability. Yet it is common for people to speak as if they are. Whenever I am contacted by the press with a question about the lottery or some other gambling game, I frequently encounter the phrase "What are the odds of ...?" In fact if I tell them that the chances of a particular event are 1 in 5 they will often quote me as stating that the odds are 5 to 1. Wrong!

To sort all this out let me use two separate notions: "odds" and "true odds." Odds, as I am using the term, refers to a payout function. To illustrate this let's consider a Place bet on the 8 in the game of Craps. The odds on this wager are 7 to 6. What this means is that you have to wager 6 units to win 7. That is, if you win you receive 13 units in all; your original 6 plus 7 more. If you lose you simply lose your 6 units. These odds, however, do not reflect the probability of winning or losing the bet. If you place the 8 the only numbers that matter are the 7 and the 8. There are 6 ways to make the 7 and 5 ways to make the 8 so a fair payoff function would pay odds of 6 to 5. This is what I, and other writers, refer to as true odds. True odds are intimately tied up with the probability of winning and losing.

If p is the probability of winning a wager and q is the probability of losing it, then the fraction q/p reduced to lowest terms, say a/b, gives true odds of a to b. In the above example, the probability of winning is 5/11 and the probability of losing is 6/11 so dividing 5/11 into 6/11 yields 6/5 or true odds of 6 to 5.

Here is another example. In sports betting the bookies offer a point spread. The purpose of this is to produce a wager that has true odds of 1 to 1. Whether or not the point spread achieves this purpose is another matter. Bookies overcome this possibility by laying off bets on one another so that they have approximately as many wagers on one team as the other (which is what the point spread is trying to achieve). The actual odds are typically 11 to 10, meaning that you have to wager 11 units to win 10. On half the bets the bookie takes in 11 units and on the other half he pays out 10 units. The house edge is

H.E. = (1/2) x 11 - (1/2) x 10 = ½

which on a per unit bet basis is 1/22 or approximately 4.545%.

Now here is a question for you. When I told the reporter that the chances of winning were 1 in 5 what should she have reported as the true odds? See you next month.

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

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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