Can an Outcome’s Cumulative Probabilities Exceed 1?
Only in Vegas!
Published online: 28 December 2017
International Atlantic Economic Society 2017
A probability distribution refers to the way in which the total probability of 1 (or 100%)
is distributed over various possible outcomes. A discrete probability distribution is
characterized by outcomes that are countable and limited. The most common applica-
tions include flipping a coin and rolling dice. Even bookmakers’ odds, when converted
to probabilities, can serve as an example. According to the opening odds from the
Westgate Las Vegas SportsBook, the odds of the Chicago Cubs winning the 2017
World Series were 3-to-1 (www.espn.com). By dividing 1 by (3 + 1), we can derive a
probability of 0.25 (25%) that the Cubs will win out. Other teams with relatively short
odds were the Los Angeles Dodgers, the Washington Nationals and the Boston Red
Sox (all 10-to-1 or 0.0909). The lowest-rated clubs were the Arizona Diamondbacks,
the Milwaukee Brewers, the Minnesota Twins, the Los Angeles Angels, the Atlanta
Braves, the Oakland Athletics, the San Diego Padres, the Cincinnati Reds and the
Philadelphia Phillies (all 100-to-1 or 0.0099).
What separates bookmakers from statisticians is the rule that cumulative probabil-
ities must equal 1. When we converted Westgate’s odds to probabilities for the 30
baseball teams in 2017, the sum was 1.3848. The unfair but intended result is that the
odds are understated. Longer odds would produce lower probabilities and a lower sum
as well as larger payouts for bettors. Reconciling baseball’s implied probabilities with a
sum of 1 could start by lengthening the odds for clubs with virtually no chance of
winning the World Series. However, if the odds for the nine clubs with odds of 100-to-1
were lengthened to, say, 1000-to-1, the cumulative probabilities would only shrink by
0.0801 to 1.3047.
Atl Econ J (2018) 46:135–136
* Ladd Kochman
Coles College of Business, Kennesaw State University, Kennesaw, GA, USA