Using Poisson Distributions for Football Prop

A few years before his 1840 death, renowned French mathematician Siméon Poisson published a distribution model now commonly used for pricing football proposition bets. Unfortunately for him, his death came half a decade before football was invented and he never got to experience the internet or online betting. Thankfully for us, his work has been well preserved and is the football betting strategy I'll discuss in this article: pricing football props using Poisson distributions.

Total Sacks Prop Bet

The Challenge: We're shopping the over/under prop betting odds on total sacks in the Giants/Redskins game and see the following the odds offered.

Topbet o4.5 +170 / u5.0 -210

BetOnline o4.5 +170 / u5.0 -210

5Dimes (1): o4.0 -110 / u4.0 -110

5Dimes (2): o5.5 +340 / u5.5 -390

Pinnacle: o4.0 -108 / u4.0 -108

We're in a gambling mood! Even though we understand handicapping the market doesn't always work well for football props, we're going to choose to give the market credit. What we know is that 5Dimes and Pinnacle are reduced juice sportsbooks while the other two are recreational betting sites mostly focused on sports betting bonuses. With this we conclude that it would appear the fair market price is a coin flip on over or under four sacks (+100 / +100). Knowing this: is there any value in the 5Dimes alternate line of 5.5 or in the 4.5 line two other sites are offering? We can figure this out using Poisson Distributions.

Solving Poisson Distributions in Excel

If you have Microsoft Excel, Poisson is easy to calculate using the Excel formula function:

=POISSON(x, mu, cumulative)

X is the number we're solving for (which we'll need to run for 4.5 and 5.5)

Mu is our calculated expectation (in this case 4)

Cumulative is asking whether or not we're solving for a range. Here we are, so we enter "true". If we we're looking for an exact probability of a specific outcome we'd enter false (for example exactly 5 sacks).

Solving Our Example

Knowing the expectation is 4, to solve for 4.5 we head to Excel, pick any cell and enter:

=POISSON(4.5, 4, true). That cell now displays 0.628837, which is 62.29%. Go to our odds converter and enter in 62.29% in the implied probability field and see in American odds format this is -165. We now know if 4 is the true even money price, then the fair prices on 4.5 is o4.5 +165 / u4.5 -165. If the market pricing of 4 being even money is correct, betting o4.5+170 at either Topbet or JustBet would be a +EV bet.

To solve for the 5.5 we enter =POISSON(5.5, 4, true) and see the solution is 78.51%. Our odds converter tells us this is -365. So a no-vig line would be o5.5 +365 / u5.5 -365. Seeing as 5Dimes had this as o5.5 +340 / u5.5 -390 neither side is +EV so we end up instead making the o4.5 +170 bet solved for above.

Dangerous Warning About Using Poisson

In order to catch your attention, I started this article by quickly showing how to use and calculate Poisson distributions before explaining what they actually are. It's extremely important I note Poisson is not appropriate for pricing most prop bets. For example, if you were pricing how many rushing yards a player would have - this would give you a very out-of-whack figure and cause you to make -EV bets. With this said let me explain what a Poisson distribution is.

What is a Poisson distribution?

In layman terms, a Poisson distribution is basically a method of quantifying the probability of random occurrences over a time period. It can only be used if an average number of occurrences over a period of time are known, and each occurrence is entirely independent of one another. Poisson are most accurate when the expected number of occurrences is small while the opportunity for an occurrence is large. Finally the average number of occurrence must be proportional, meaning if the time interval doubled so would the number of occurrences expected.

Back to my point about not being able to use Poisson for a player's rushing yards. This lacks the randomness element; as mentioned, each occurrence must be independent of the last. A player rushing one yard is more likely to take steps and rush another. The events need to occur one at a time, with complete independence. With this all said, we can now make a criteria list:

The opportunity for an occurrence must be large.
The actual number of occurrences must be small.
Occurrence must happen one at a time.
Each occurrence must be independent and random.
Number of occurrences over a time period (meaning, if the time period doubled so would the expected number of occurrences).

These 5 criteria eliminate using Poisson distribution for all sorts of bet pricing. As we established it can't be used for rushing yards, and likewise can't be used for passing yards. It also can't be used for number of completed passes as these occur far too frequently per attempt. Scoring is off the list as well for football, and to note would be for hockey as well due to failing the proportional test. In hockey doubling the outcome would not lead to the same expected occurrence due to teams pulling their goalie at the end of games. To show where it does work, allow me to break down pricing another prop.

Number of Interceptions Prop Bet

For the wild card round of the 2011/2012 NFL playoffs, Detroit Lions @ New Orleans Saints, I came across a prop bet total interceptions by both teams:

o1.5 -120 / u1.5 +100

This bet caught my eye because normally I see this priced much higher, such as o1.5 -160 / u1.5 +130 or something similar. My natural inclination was to immediately bet the over. My thought process was this is two passing teams, the game has a high betting total, and so there is a ton of chances for INTs. Of course, as someone who bets sports for a living, I quickly realized I was thinking like a football betting square (a novice bettor). I decided to investigate further.

My first step was to head to www.nfl.com pull the stats on season interceptions for each QB, each defense, and all defenses. From here I broke the stats down to per game averages as follows:

Drew Brees: 0.875 per game

Matthew Stafford: 1 per game

Lions: 1.313 per game

Saints: 0.563 per game

League Average: 0.988 per game

In order to calculate a QB's expected interceptions per game, we need to reconcile his figures against his opponent's defense. However, NFL seasons are short. With just 16 games in a season, it's important we normalize the defenses data by incorporating league average into our equation. The formula I use that works quite well for this is:

(QB INT Per Game)*(Defenses INT per Game / League Average INT Per Game) = Expected INTs.

Using this above formula I get:

Brees: 0.875*(1.313/0.988)= 1.163

Stafford: 1*(0.563/0.988)= 0.570

Total Expected for Game: 1.733

From here I need to ask myself a few questions to see if pricing this prop meets the Poisson distribution model criteria.

Is there a high number of potential occurrences? Looking over season stats, each of these of these quarterbacks has averaged just over 41 pass attempts per game. This gives me 82.3 expected trials. It's not a huge number, but I'm confident that it's close enough to give me a fairly accurate estimate.
Are the number of expected occurrences small? 1.733 expected occurrences / 82.4 expected trials = 2.1%. Again, while a little higher than I'd like, it is low enough to give me a fairly decent estimate.
Do the occurrences happen one at a time? Most definitely. A QB can't throw two interceptions on the same passing attempt.
Are the occurrences independent of one another? For all intents and purposes, yes. However, one could argue a quarterback going on mental tilt could throw more interceptions out of frustration. Point noted, but again this should be close enough.
If the game's length was extended would the number of occurrences remain proportional? Without stretching it to fatigue, the answer is… yes, they're proportional.

My conclusion here is Poisson won't be so accurate that I can bet a 1% edge, but is accurate enough that if the edge is large, I can make a bet. So my next step was opening Excel and plugging in = =POISSON(1.5, 1.733, TRUE) to see the chances of going under 1.5 are 48.3%. Using my odds converter I was able to price this prop at o1.5 -107 / u1.5 +107.

This was a case where knowing Poisson distributions saved me from making a snap reaction -EV bet. As I watched the game head into the final 10 minutes with no interceptions thrown, I felt great about having saved a bet. However, it turned out on this day math was my detriment. Late in game Stafford was picked off twice, one of them coming in garbage time. This is fine; to beat props long term it's far better to do so by making +EV bets and avoiding -EV bets, than it is to get lucky.

Best Use of Poisson Distribution

For starters, I suggest reading my main article on football prop betting. Although it does not cover handicapping props that Poisson distribution can be used, it does give more insights into prop pricing. From here, once you're able to crack handicapping props that can be priced with Poisson distributions, you'll want to have an account at www.bovoda.lv. This is because Bovada makes a large number of line shades, for example when the market on total field goals is o3.0 -120 / u3.0 -110, Bovoda will often offer a line on 3.5. Knowing Poisson and shopping props at Bovada is a great way to make money prop betting long term.