The Overround

NFL Win Totals are usually presented thus:

Chicago being convened with a total as an integer, and Arizona given a half point, allowing for a push rate of zero.

Simply, to come up with an expected winning percentage, one could just take the total number and divide by 16 (total regular season NFL games). This equates to an expected winning percentage for Chicago of 50%, and Arizona 46.88%.

But that’s a rather banal and uninviting number when being given the odds of each event happening. I’ve discussed how to convert odds to win probability, then from there to fair value win probability, and feel free to use my calculator to that end.

After conversion the table is reconfigured like so:

The table is saying the Chicago Bears have a 46.3% chance of winning more than 8 games, and 53.7% of losing less than 8 games. Similarly for the Arizona Cardinals, who have been appropriated with a 56.02% probability of going over 7.5, and 43.98% of winning 7 or fewer.

What we have now is the elements required for a binomial probability scenario. A probability of a certain number of events of one variable resulting in success or failure given a sample size, and the success probability of one single event. In this case the sample size, n is 16, and the number of successes, x is the win total. What is missing is the precise measure of success in any one game. But what is known is the answer to the cumulative binomial distribution equation. And that is the fair value odds.

One condition that demands further attention before calculating is the push probability of the 8 wins, for Chicago, that Pinnacle is showing as the win total. The Bears could very well win 8 victories, which must be accounted for. Before I proceed its necessary to remove the push probability from the equation.

I’ve introduced the framework of a binomial distribution and what that entails here, and explained to capacity, so I won’t expound further. I highly suggest not only reading my post but seeking information at Wikipedia here, and a site aimed at explaining Binomial Probability here. Both offer far more worthy and articulate explanations of the concepts involved, than what one may gather from my elaborate drivel.

To Wit:

given probability p, sample size n, number of successes x

Probability Mass Function

Cumulative Distribution Function

For the latter, the variable i is all successes up to x. The probability that any value less than or equal to x number of successes in sample size n.

The fair value over price is the cumulative Probability, P, of winning more than x in a 16 game season. While the formula provides the “floor” of success, x, to find the other end of the spectrum subtract the resulting number from 1.

The former equation, for convenience I’ll refer to as PMF, is used to determine the possibility of the Bears winning 8 games, which is the push rate.  Again Pinnacle provides us with the answer to the CDF equation.  The Bears fair value over percentage is 46.30%.  The known variables:

P(X=x) = 46.30%
x= 8
n = 16

Stata has a built in function to calculate the binomial probability with the single game probability unknown, yet the answer to the formula assumed. For those afforded the luxury of having Stata at their immediate disposal, the command “invbinomialtail (n,x,P)” renders the answer to the single game probability.

Without Stata, and in avoidance to having to go through the trouble of solving the equation for the variable p, the Excel Solver add-in serves as a viable alternative.

Copy the data for the Chicago Bears into corresponding cells in excel. Calculate the winning percentage for the Bears, preferably just enter =”8/16″ in its appropriate cell. Then in a cell adjacent, place this formula:

“=1-binomdist(wins-1,16,probability,true)”

Now open solver and set the target cell to the one containing the formula above, a minimum value of 0, by changing the organic winning percentage (total / 16), and the solution constrained to the value equal to the fair value over odds.

Copy the altered winning percentage to an unoccupied cell. Enter the formula into another adjacent cell:

“=binomdist(wins,16,probability,true)”

Run solver again, except this time set the solver result equal to the fair value under odds.

Average both winning percentages derived from the above equations.

For the Bears their projected winning percentage after negotiating the operation described above is 48.86%, or 7.82 wins.  Now enter the probability, p, of 48.86% into the PMF equation, and the push rate ≈ 19.49%.

Future win totals with a half point, such as the Cardinals, are far more accessible. Little effort is required after purveying the aforementioned processed. As a way of double checking the validity of the projected single game winning percentages, the sum total of all the new win totals should be roughly 256. 256 is the maximum number of wins distributed through the course of the NFL season. And after arbitrarily giving the currently OTB Minnesota Vikings 8 wins, the sum of all the win totals is equal to 256.79. Conversely, had you not taken the push probability into consideration, the sum total would have been 234.

To speed up the calculations, I have written an excel macro to run solver equations in a loop. Solver uses absolute references, disabling the ability to use the cell range (i.e. “…Cells(a,b).Value”) in the Macro. Its an easy fix, by way of cell address and the looped variables i and k.

Just point the code below to whatever cells contain the relevant data.

Sub NFL_WPSolver()
On Error Resume Next
Application.Calculation = xlCalculationManual
Application.ScreenUpdating = False
Dim i As Integer
Dim k As Integer
Dim Lastrow As Integer
i = 1
k = 1
Lastrow = Range("teams").Rows.Count
For i = 1 To Lastrow
    SolvReset
    SolverOk SetCell:="$L$" & i + 1, MaxMinVal:=2, ValueOf:="0", ByChange:="$J$" & i + 1
    SolverAdd CellRef:="$H$" & i + 1, Relation:=2, FormulaText:="$L$" & i + 1
    SolverOk SetCell:="$L$" & i + 1, MaxMinVal:=2, ValueOf:="0", ByChange:="$J$" & i + 1
    SolverSolve True
Next i
For k = 1 To Range("teams").Rows.Count
    SolvReset
    SolverOk SetCell:="$M$" & k + 1, MaxMinVal:=2, ValueOf:="0", ByChange:="$K$" & k + 1
    SolverAdd CellRef:="$I$" & k + 1, Relation:=2, FormulaText:="$M$" & k + 1
    SolverOk SetCell:="$M$" & k + 1, MaxMinVal:=2, ValueOf:="0", ByChange:="$K$" & k + 1
    SolverSolve True
Next k
Application.Calculation = xlCalculationAutomatic
Application.ScreenUpdating = True
End Sub

Of consequence is the essence of what the concluding numbers represent. Logistically its the respective team’s average single game winning percentage over the course of the season. When comparing two teams using these win total projections, to find the expected Vegas Moneyline, just insert the team single game win probability into the Log 5 calculator.

From there one can take it a step further and find the expected line based on some of the newly constructed information, which entails a slight inversion of the Pythag formula. I’ll explore those contingencies at a later date.