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.

The Poisson distribution model was published
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.

  1. The opportunity for an occurrence must be large.
  2. The actual number of occurrences must be small.
  3. Occurrence must happen one at a time.
  4. Each occurrence must be independent and random.
  5. 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.

Bookmaker A
Total Sacks: 4.5

Bookmaker B
Total Sacks: 4.0

Bookmaker C
Total Sacks: 5.5

Bookmaker D
Total Sacks: 5.5

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

Excel 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.

Excel Function

=POISSON(4.5, 4, true)

That cell now displays 0.628837, which we convert to a
percentage of 62.29%. Go to our odds
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

To solve for the 5.5 we enter the following.

Excel Function

=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

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

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.

Excel Function

=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.

Beating props long term requires making +EV bets and avoiding
-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.