Obtaining Estimates of the Probability of Winning From the Baseball Betting Market

Follow a 3-step procedure

1. Translate the "money line" in the newspaper to odds
2. Calculate p^FAIR = 1/(1+Odds) for each team
3. Normalize so that the probabilities sum to 1.0, i.e. can be thought of as "true" probabilities: p^FAIR* = p^FAIR / sum(p^FAIR)

Example: 9/29/99 Reds Vs Astros

Notes: The money line is determined by the market.

Step 1: translate into odd to $1

+140 Indicates the a win nets $140 for every $100 bet

Since you win $1.40 for every $1 (if successful), the Odds Equivalent is 1.4/1.

-150 indicates it takes a bet of $150 to net $100

Since you win $1 for every $1.50 bet (if successful), the Odds Equivalent is 1/1.5 or 0.67.

Step 2: use the 1/(1+Odds) rule to calculate p^FAIR

Step 3: divide the numbers in the next to last column by the sum (1.0167) to obtain the normalized probabilities

Why are these "good" estimates of the probabilities?

1. Given the odds, ER = p^FAIR* (1 + Odds) < 1

That is, the expected return to a $1 bet does not exceed $1. On average, you would not earn a profit, which would be the case if ER > $1. We don't expect the bookmaker market to consistently offer bettors the opportunity to get rich at the bookmakers' expense.

2. Prior to normalizing, we calculated the unadjusted p^FAIR by setting ER = 1. But this is just an intermediate step, and we know the initial values of p^FAIR must be adjusted downward since their sum exceeds 1. True probabilities can't have a sum >greater than 1.

3. Hence we must reduce the sum, and normalizing by the sum is an efficient way to do this which is consistent with economic theory (equilibrium odds exhibiting the lack of profit opportunities).