Calculating Probabilities with Negative Binomial

Now that you know when to use Negative Binomial and how to estimate r, let's calculate probabilities. The process is similar to Poisson (Chapter 8), but with an extra step to calculate the p parameter that Excel requires.

Excel's NEGBINOM.DIST Function

NEGBINOM.DIST Syntax

=NEGBINOM.DIST(k, r, p, cumulative)

Excel: k=successes, r=dispersion, p=r/(r+μ), cumulative=FALSE/TRUE

Parameter Breakdown

Parameter	Meaning	How to Calculate
k	Number of successes (e.g., 2 touchdowns)	The count you're evaluating
r	Dispersion parameter	From method of moments: μ²/(Variance-μ)
p	Probability parameter	p = r / (r + μ)
cumulative	FALSE = exact, TRUE = cumulative	FALSE for P(X=k), TRUE for P(X≤k)

Warning

The p parameter can be confusing—it's not the probability of success in the intuitive sense. It's a mathematical transformation that Excel requires to parameterize the distribution. Just remember: p = r / (r + μ), and you'll be fine.

Common Excel Patterns

Probability	Excel Formula
Exactly k	`=NEGBINOM.DIST(k, r, p, FALSE)`
k or fewer	`=NEGBINOM.DIST(k, r, p, TRUE)`
k or more	`=1 - NEGBINOM.DIST(k-1, r, p, TRUE)`

Complete Worked Example: Backup Running Back TD Prop

Let's walk through a complete analysis for a backup running back who gets goal-line carries but has inconsistent opportunities.

The Market Line

Side	Odds	Implied Probability
Over 0.5 rushing TDs	-130	56.5%
Under 0.5 rushing TDs	+105	48.8%

Your Model Data

From 20 games of historical data:

Metric	Value
Historical Mean (μ)	0.85
Historical Variance	1.03
VMR	1.21
Calculated r	4.00

Matchup Adjustment: Facing weak run defense → increase μ to 0.95

Final parameters: μ = 0.95, r = 4.00

Step 1: Calculate p

p = r / (r + μ)
p = 4.00 / (4.00 + 0.95)
p = 4.00 / 4.95
p = 0.808

Step 2: Calculate P(0 TDs) for Under 0.5

=NEGBINOM.DIST(0, 4.00, 0.808, FALSE)

Result: 0.436 or 43.6%

Step 3: Compare to Poisson

If we had used Poisson with μ = 0.95:

=POISSON.DIST(0, 0.95, FALSE)

Result: 0.387 or 38.7%

Key Insight

The Difference: Negative Binomial gives 43.6% probability of 0 TDs, while Poisson gives only 38.7%. That's a 4.9 percentage point difference. The Negative Binomial accounts for the player's boom-or-bust nature, assigning higher probability to the extreme outcome of a shutout. This is the fatter tail in action.

Step 4: Compare to Market (Under 0.5)

Metric	Value
Under 0.5 at +105	Implied P = 48.8%
Your Negative Binomial P	43.6%
Market edge	43.6% - 48.8% = -5.2%

The Under is overpriced by the market. No bet on Under.

Step 5: Calculate P(1+ TDs) for Over 0.5

P(1+ TDs) = 1 - P(0 TDs) = 1 - 0.436 = 0.564 or 56.4%

Step 6: Compare to Market (Over 0.5)

Metric	Value
Over 0.5 at -130	Implied P = 56.5%
Your Negative Binomial P	56.4%
Your edge	56.4% - 56.5% = -0.1%

Note

Conclusion: Both sides are fairly priced. The market appears to be accounting for the overdispersion already. This is a sharp line—no bet.

If you had used Poisson (which gives 61.3% for 1+ TDs), you might have thought the Over at -130 was a good bet. But Negative Binomial reveals there's no edge. This demonstrates why detecting overdispersion matters—it prevents you from making -EV bets.

Building a Negative Binomial Calculator in Excel

Set up your spreadsheet with these columns:

Column	Header	Content/Formula
A	Player	Name
B	Historical Mean	From data
C	Historical Variance	From data
D	VMR	`=C2/B2`
E	r (Dispersion)	`=B2^2/(C2-B2)`
F	Matchup Adj.	Multiplier (e.g., 1.10)
G	Adjusted μ	`=B2*F2`
H	p Parameter	`=E2/(E2+G2)`
I	Line	The prop line (e.g., 0.5)
J	Odds	American odds (e.g., -130)
K	P(Over)	`=1-NEGBINOM.DIST(INT(I2),E2,H2,TRUE)`
L	Market Implied	`=IF(J2<0,ABS(J2)/(ABS(J2)+100),100/(J2+100))`
M	Edge	`=K2-L2`

Multi-Outcome Example: 2+ Touchdowns

Sometimes you want to bet on multiple touchdowns at longer odds. Let's calculate P(2+ TDs) for our backup RB:

Parameters: μ = 0.95, r = 4.00, p = 0.808

P(2+ TDs) = 1 - NEGBINOM.DIST(1, 4.00, 0.808, TRUE)

Result: 19.8%

Compare to Poisson:

P(2+ TDs) = 1 - POISSON.DIST(1, 0.95, TRUE)

Poisson Result: 22.5%

Warning

Wait—Poisson gives a higher probability for 2+ TDs? Yes, because the Negative Binomial's fatter left tail (more probability at 0) takes probability mass away from the middle outcomes (1-2 TDs) and redistributes it to both extremes (0 and 3+).

For this specific example:

P(0): NB is higher (43.6% vs 38.7%)
P(1): NB is lower
P(2): NB is slightly lower
P(3+): NB is higher

The crossover point depends on μ and r.

Negative Binomial Calculator

Try the interactive calculator for this concept

Open Tool

📝 Exercise

Instructions

A boom-or-bust receiver has μ = 0.75 TDs, r = 1.5. The market offers Over 0.5 TDs at -115.

Calculate:

The p parameter
P(0 TDs)
P(1+ TDs)
The market implied probability
Your edge (if any)