# Using Poisson Distributions for Football Props

In our article covering the

basics of betting football props, we explained how football

props are widely considered to be better suited to recreational

bettors. This is because they are largely luck-based wagers. We

also explained that it’s possible to handicap certain types of

football props, and that they shouldn’t be dismissed by bettors

serious about trying to make money.

The key to making **profitable football prop bets** is ultimately

in learning how to price them better than the bookmakers do.

This is how you find value in them, and it’s an art that we

explain more in the article linked above.

In this article we focus on a specific method that can be

applied for pricing props – **using Poisson distributions**. We

explain what Poisson distributions are, and illustrate how they

can be used when betting on football.

## 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 distributions are most accurate when the expected

number of occurrences is small while the opportunity for an

occurrence is large. The average number of occurrences must also

be proportional, meaning if the time interval doubled so would

the number of occurrences expected.

by the renowned French mathematician Simeon Poisson, back in the

19th century.

Unfortunately for the inventor of this model, he died 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 his model is used today in a number of

aspects of sports betting.

## Using Poisson Distributions: A Warning

It’s extremely important to note that this distribution model

isn’t 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. They lack the randomness element required where

each occurrence must be independent of the last. A player

rushing one yard is more likely to take steps and rush another.

The following is a list of criteria that must exist in a prop

bet for the Poisson distribution model to work effectively.

- 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 five criteria eliminate using Poisson distribution for

all sorts of bet pricing. As we established, it can’t be used

for rushing yards due to the lack of randomness and the need for

events to occur one at a time. It can’t be used for passing

yards for the same reasons. It also can’t be used for number of

completed passes, as these occur far too frequently per attempt.

Scoring is off the list for football due to failing the

proportional test.

To show where it does work, we’ll illustrate it’s use in two

specific types of football prop bet.

## Using Poisson for Total Sacks Prop Bets

Let’s say we’re shopping the over/under prop betting odds on

total sacks in a game between the Giants and the Redskins. We

see the following odds offered.

Total Sacks: 4.5

+170

-180

Total Sacks: 4.0

-110

-110

Total Sacks: 5.5

+340

-390

Total Sacks: 5.5

-108

-108

Even though we understand that handicapping the market

doesn’t always work well for football props, we’re going to

choose to give the market credit. Based on our knowledge of the

different bookmakers, and which ones are for recreational

bettors and which ones are offering reduced juice, we conclude

that it would appear the fair market price is around even money

on over or under four sacks.

The goal is now to determine whether there is any value in

the lines of 4.5 and 5.5 being offered. We can figure this out

using Poisson Distributions. The easiest way to do this is to

use Excel, as you just need to use the following 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 were looking

for an exact probability of a specific outcome we’d enter

false (for example exactly 5 sacks).

Knowing the expectation is 4, to solve for 4.5 we head to

Excel, pick any cell and enter the following.

=POISSON(4.5, 4, true)

That cell now displays 0.628837, which we convert to a

percentage of 62.29%. Go to our odds

converter and enter in 62.29% in the implied probability

field and you see in American odds format this is -165. If we’re

assuming that 4 is the true even money line, then the fair

prices on over/under 4.5 are +165 for the over and -165 for the

under. This would mean betting the over 4.5+ at +170 with

Bookmaker A is a +EV wager, as the odds are better than the fair

price.

To solve for the 5.5 we enter the following.

=POISSON(5.5, 4, true)

The solution converts to a percentage of 78.51%. Our odds

converter tells us this is -365. So a no-vig line would be over

5.5 at +365 and under 5.5 at -365. Seeing as the bookmaker

offering the 5.5 line has odds of +340 and -390 respectively,

neither side is +EV as the odds are worse than the fair price.

## Using Poisson for Total Interceptions Prop Bets

Here we’re going to use an example of a real betting line

found by one of our team back during the 2011/2012 playoffs. It

was for the total number of interceptions in the game between

the Detroit Lions and the New Orleans Saints. The total was set

at 1.5, with the over available at -120 and the under at +100.

This bet caught the eye because normally it is seen priced

much higher, such as -160 for the over and +130 for the under.

The natural inclination was therefore to immediately bet the

over. With two passing teams, and a high betting total, there’s

going to be a ton of chances for interceptions. Being a

professional bettor, our guy was not just going to trust his

instinct so easily though. He investigated further.

His first step was to head to pull the stats on season

interceptions for each QB, each defense, and all defenses. From

here he 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

Calculating a QB’s expected interceptions per game means

reconciling his figures against his opponent’s defense. However,

NFL seasons are short. With just 16 games in a season, it’s

important to normalize the defense data by incorporating league

average into the equation. A formula that works quite well for

this is as follows.

Using the above formula resulted in the following expected

interceptions for each quarterback.

- Brees: 1.163
- Stafford: 0.570

Adding these two together gives the expected total for the

game, which is 1.733.

From here, the next step was to see if this prop met the

Poisson distribution model criteria.

#### Is there a high number of potential occurrences?

Looking over season stats, each of these quarterbacks has averaged

just over 41 pass attempts per game. This gives 82.3 expected

trials. It’s not a huge number, but close enough to give 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 is ideal, it’s low enough to

give 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.

The conclusion here is that Poisson won’t be so accurate that

a wager could be placed with a low edge, but it’s accurate

enough that a wager could be placed with a large edge. So the

next step was opening up Excel and plugging in the following.

=POISSON(1.5, 1.733, TRUE)

This was solved to suggest that the chances of going under

1.5 were 48.3%. Using our odds converter, the fair prices came

out at -107 for the over and +107 for the under.

This was a case where knowing Poisson distributions saved an

experienced and knowledgeable bettor from making a snap reaction

-EV bet. As the game headed into the final ten minutes with no

interceptions thrown, our guy no doubt felt great about having

saved some money.

As it turned out, however, math was to his detriment that

day. Late in game Stafford was picked off twice, one of them

coming in garbage time. This highlights a very important point

though, and a perfect one to finish on.

-EV bets, not getting lucky.

On the occasion described above, a –EV bet would have won.

That happens sometimes. Equally, +EV bets lose sometimes. But

the whole concept of expected value is based on the long run. If

you consistently make +EV wagers then you should make profits

over time. And Poisson distributions can help you to do exactly

that with football prop bets.