Thanks to late **French Mathematician Simon Denis Poisson**, we’re able to calculate the probability of sports betting wagers using Poisson distribution. Poisson published his works in the 19^{th} century, but they still work perfectly to determine the probability of each possible outcome.

His formula allows modern bettors to examine each side of a betting proposition and determine their edge, or lack thereof.

**Poisson and Binomial Distributions**

Many propositions fall under the category of Poisson while others firmly fall under binomial distribution. Understanding the difference is crucial to finding +EV opportunities when betting propositions.

## Is it a Poisson Distribution?

There are **two main factors** that determine if an event or proposition falls under Poisson distribution. Firstly, the variable that players are measuring for **must be counted at one time**. Secondly, the probability of the event you’re **calculating must be small**, but the **number of chances** to achieve the event **has to be large**.

**Also….**

The event also has to be proportional to the time period. For instance, if we calculated the number of penalties in 48 minute basketball game, we would expect there to be twice as many penalties if the contest were 96 minutes long.

Points scored falls under Poisson, but only if the amount scored is one point. However, Poisson is best used on wagers such as the number of turnovers in an NFL game or the number of sacks. These numbers would be totaled from both teams. Players can use historical data to form their own historical averages to calculate their chances against the bookmaker’s line.

Using Poisson, there’s only one possible outcome when the event occurs. If a quarterback gets tackled behind his line of scrimmage, he goes down for a sack. This is counted as 1.0 sack in the box score for the game regardless of which QB has been brought down.

Binomial Bets: On the flip side, binomial distribution applies when the event has two possible results.

**Example:** The most basic example of this is a coin toss, which has two possible outcomes, heads or tails.

It’s important to recognize the difference between binomial and Poisson distributions when examining your edge. Binomial distribution can also begin to resemble Poisson if one of the outcomes is extremely rare, but still has a large number of possible occurrences.

## Examples of Poisson

Similar to the above mentioned prop regarding the number of sacks in an NFL game, we will go over the number of turnovers per contest. Like sacks, turnovers are recorded as a single unit when they happen. They also have a small probability of happening on every play, but have a large number of potential occurrences.

The best way to calculate Poisson is through Microsoft Excel. In Excel, Poisson can be solved by using the Excel formula function by entering:

**=POISSON(x, mean, cumulative). **

Without the help of Excel, bettors would be forced to use a statistical database to find the Poisson probabilities.

**Total Turnovers**

Sample odds at several of the top sportsbooks offer many different prices for the same prop:

**5Dimes: **o4.0 -110/ u4.0 -110

**Pinnacle:** o4.0 -107/ u4.0 -107

**BetOnline:** o5.0 +210 / u5.0 –230

**TopBet: **o4.5 +180/ u4.5 -200

## =POISSON(x, mean, cumulative)

**X**: The number we’ll be solving for; in this case the numbers are 4.0 and 5.0.

**Mean**: Our expectation, for this example we’ll go with an average of 4.0 turnovers per game. This number would usually be based on historical data or averages, but we’re just using an example.

**Cumulative**: Asks if we are solving for a range of data. For this example, we would enter ‘true’ because we’re not looking for an exact outcome.

If we calculate for 4.0, our formula is =POISSON (4.0, 4.0, TRUE), which comes out to 0.628837 or about 62.88 percent. This is our implied probability, which can be converted into American odds easily through an odds converter or by doing our own math. The calculated conversion is -169.41, so a fair market price on the over or under at 4.0 is around -169.

Our calculations eliminate 5Dimes’s and Pinnacle’s odds from our consideration because our breakeven odds of -169 is far from their pricing range of -107 to -110 on both sides of the bet. However, TopBet’s o4.5 +180 odds offer an excellent value and +EV betting opportunity. Not only are we beating our breakeven odds, but we’re gaining a half point.

When we calculate for 5.0, our formula is =POISSON (5.0, 4.0, TRUE), which comes to 0.78513, or 78.51 percent. After we convert this to American odds, a no-vig moneyline on this bet comes to around -365.

This calculation offers us little value because our breakeven odds would need to be +365 / -365. None of the above examples are even close to this range, so we’ll pass.

**Using Poisson Calculations**

Using Poisson distribution to determine value in betting markets, especially for propositions is a strategy that all bettors should employ. As I have mentioned in other articles here at OnlineBetting.com, derivatives and smaller markets are a perfect target for newer bettors to build a bankroll.

However, bettors can get themselves caught into the trap of calculating Poisson distribution for bets that aren’t Poisson. These include wagers such as a running back’s rushing yards, a receiver’s reception yards or a basketball player’s points + rebounds. This can lead to making some potentially negative expectation bets.

Choose your spots wisely and be sure to follow the appropriate guidelines for identifying Poisson opportunities. Using the distribution is one of the most effective tools for spotting inefficiencies and finding value in betting markets.

**Authors and Contributors: Joseph Falchetti & Jim Griffin**