Sports Betting Probabilities

Very common to sports betting propositions are the challenges of calculating the number of occurrences of an event during a game, such as the number (but not the length) of punts by one team, or shots taken (but not points scored) by a particular player. Individual counts are treated somewhat differently from those statistics relating to accumulations, like points, yardage, ERA's or batting averages. The Poisson Distribution describes the probability of having a certain number of "hits" during a fixed (and standardized) period, in which the event in question has a relatively low probability of occurrence, but a large number of chances of occurring. These chances would be shots taken or plays in a football game, pitches in baseball, etc.

For example, if the question relates to quarterback sacks, and the handicapping shows that a specific team's offensive line gives up an average of 3 per game, what is the possibility that there will be fewer than 3 sacks in this game? The cumulative Poisson Distribution tables show that for a mean of 3, the probability of only 0, 1 or 2 sacks is 42%. Thus, the probability of having 3 or more sacks is 58%. If a -110 prop on sacks states 2½ sacks (so that 3 is a winner), it is a good bet. (The expectation is $210*.58 or $121.80. The profit of $21.80 on a bet of $110 comes to 19.8%.) If the proposition were to state 3 sacks (so that 3 is a push), the calculation becomes $210*.42 + $110*.22+$0*.36 or $112.40, just barely over the cost to play. The "profit" would be $2.40 or 2.2%.

The Poisson Distribution will be helpful in the comparison of two "integer" events of the appropriate kind. Suppose, for example, that the proposition is that there will be not be any more quarterback sacks against the home team in a certain game than against the away team. This kind of proposition compares two distributions, and is subject to the random error of both. Assume the home team gives up, on average, 3 sacks per game, but the away team gives up only 2 sacks per game on average. According to a two variable Poisson table, the probability that the home team will give up more sacks than the away team is 58%. There is a 25% chance that the away team will give up more sacks than the home team, and a 17% chance that they will have the same number. At -110, this proposition is a good bet.

Poisson tables are widely available. Using them is a significant benefit in analyzing "integer" propositions - those where the outcome in question is a countable number of units. Think "a small number of 'successes' in a large number of 'trials'." It does not apply to quantitative measures, like points, yards, or weight. It is important always to ask first whether the proposition is of a sort where the Poisson table would help.