Odds and probabilities are important (and related) concepts in sports analytics. They appear in several contexts, such as (the related ones of) prediction, bookmaking, and betting.
Despite their importance and wide usage, there is [1] common confusion about these terms. This results in (and is compounded by) inaccurate statistical language in the dissemination of results. Information reported also often neglects subtle but important details. Understanding these issues is necessary to describe (and understand) the likeliness of events.
In this Article, odds and probabilities and their relationship are considered in detail. As an application, book odds are discussed.
Odds and Probabilities
Odds and probabilities are both statistical terms that are used to describe the likeliness of an event. How they each describe this, however, are different.
Odds
total outcomes and 
of which favor that event (and, hence, 
of which do not). Based on this definition, the odds are therefore 
where 
is used to denote against.
Probabilities
Probability can also be used to describe the likeliness of an event.
The probability of event A, 
, is expressed as
(1) 
and that not occurring is
(2) 
where 
is the complement of A.
Note that an important equality implied by Eqs. (1) and (2) is
(3) 
The Relationship Between Odds and Probabilities
Odds and probabilities are clearly related (as explicitly stated above).
In fact, they can straightforwardly be defined in terms of each other:
all other definitions (odds against, or definitions of probabilities) follow similarly.
Book Odds
In the context of sports betting, the consideration of (book) odds must be made with care. This is because there is unknown information (to the bettor) that the bookmaker has.
In making a book (the mathematics of bookmaking will be considered in a separate article), a bookmaker will reduce (from the true) odds, in order to ensure an expected profit.
Consider an example of a baseball game with moneyline odds reported as 
[away (A)]and 
[home (H)]. Converting these “odds” to “probabilities” gives
(where quotations ” and primes ‘ have been added, for the reasons to follow). Adding these “probabilities” gives
As this sums to more than one, these “probabilities” cannot be considered as (true) probabilities [and satisfy Eq. (3)].
The amount by which a book exceeds a “probability” of 
(and which represents the bookmaker’s expected profit) is known as the overround, bookmaker margin, or the vigorish (vig).
Without additional information (about the odds reduction), it is impossible to determine the bookmaker’s (true) probability estimates.
Assume, however, that the bookmaker does not want to accept a negative expected value for any bet. It can therefore be concluded that
and 
are upper bounds to the (true) probability and 
, respectively;
= 
and 
lower bounds.
Precise estimates can only be made by further assuming (or knowing) the model for reducing. Consider the simplest one, which preserves the relative “probabilities”. In this case,
where 
is the overround. Using Eq. (3), the above equation can be used to solve for , and then and 
. For the example above, this results in
References
[1] L. V. Fulton, F. A. Mendez, N. D. Bastian, and R. M. Muzal, “Confusion Between Odds and Probability, a Pandemic?” Journal of Statistics Education 20 (3) (2012)
[2] D. A. Berry and B. W. Lindgren, Statistics: Theory and Methods, 2nd ed., p. 15 (Duxbury Press, 1996)
