This example illustrates how probabilities can be assigned empirically or modeled parametrically using data from professional American football games.
It is not a Bayesian inference problem, but an example of probability assignment — how to quantify uncertainty about game outcomes using observed data and reasonable assumptions.
1. Football Point Spreads and Game Outcomes
In professional football, experts set a point spread for each game, reflecting the perceived difference in strength between two teams.
- Example: Team A is a 3.5-point favorite over Team B.
→ This means that a fair bet (50–50 odds) is that Team A wins by more than 3.5 points. - If the point spread is an integer (e.g., 3), the probability that A wins by more than 3 points equals the probability that A wins by fewer points (or loses).
→ There’s also a positive probability that A wins exactly by the point spread, in which case bets are usually a “push” (no payout).
The point spread can be thought of as the median of the betting population’s belief distribution about the game’s outcome difference.
For analysis, point spreads are treated as given, and the task is to estimate the probabilities of different outcomes based on actual data.
2. Data Overview
The dataset contains 672 professional football games from the 1981, 1983, and 1984 seasons.
(1982 is excluded due to a labor strike.)
- x = point spread (favorite’s expected winning margin)
- y = actual game outcome (favorite’s score − underdog’s score)
Each point in the scatterplot (Figure 1.1) shows a game’s (x, y) pair.
Random “jitter” (small random noise) is added to x and y to avoid overplotting since outcomes are discrete.
3. Assigning Probabilities from Observed Frequencies
We can estimate probabilities empirically by counting wins and losses from the data.
Ignoring games with x = 0 (no favorite), and counting ties as half a win, the empirical estimates are:
| Event | Empirical Probability |
|---|---|
| Favorite wins | 410.5 / 655 = 0.63 |
| Favorite wins given x = 3.5 | 36 / 59 = 0.61 |
| Favorite wins by more than the spread | 308 / 655 = 0.47 |
| Favorite wins by more than the spread given x = 3.5 | 32 / 59 = 0.54 |
These results align with general intuition:
- The favorite wins about 63% of the time.
- When favored by 3.5 points, the chance of winning is around 61%.
- The chance of beating the spread is slightly below 50%, consistent with the idea that spreads are designed for balanced betting.
However, empirical probabilities can be unstable when sample sizes are small.
For instance:
- 8.5-point favorites: 5 wins out of 5 games → 100% win rate
- 9.0-point favorites: 13 wins out of 20 → 65% win rate
We would expect 9-point favorites to have a higher true probability than 8.5-point favorites, but due to small samples, the empirical estimates are noisy.
This motivates the use of a parametric model.
4. Parametric Model for (Outcome − Point Spread)
To reduce noise and impose structure, we define:
$d = y – x$
where d is the difference between the actual margin and the point spread.
A scatterplot of (d vs x) shows no strong dependence of d on x, suggesting that d may be approximately independent of x.
A histogram of all d values (Figure 1.2b) shows that the differences are roughly normally distributed.
The sample statistics for 672 games are:
- Mean of d ≈ 0.07
- Standard deviation of d ≈ 13.86
Thus, the model assumes:
$d \mid x \sim N(0, 14^2)$
meaning:
- On average, the point spread predicts the outcome correctly (mean 0).
- The variability around that prediction is about 14 points (roughly two touchdowns).
- The distribution of d is independent of x.
This model is not perfect (since scores are discrete and the fit isn’t exact), but it provides a useful approximation.
5. Computing Probabilities from the Model
Using the model $d \sim N(0, 14^2)$, we can calculate various probabilities.
(a) Probability that the favorite wins by more than the spread
Since the mean of d is 0:
$Pr(d > 0 \mid x) = 0.5$
That is, for any spread x, there’s a 50% chance that the favorite covers the spread — consistent with the definition of the spread as a fair bet.
(b) Probability that the favorite wins the game
The favorite wins if y > 0, i.e., if $d > -x$:
$Pr(y > 0 \mid x) = Pr(d > -x \mid x) = 1 – \Phi\left(-\frac{x}{14}\right)$
where Φ is the standard normal CDF.
Hence, the model-based win probabilities are:
| Point Spread (x) | $Pr_{norm}(y > 0 \mid x)$ |
|---|---|
| 3.5 | $1 – \Phi(-3.5/14)$ = 0.60 |
| 8.5 | $1 – \Phi(-8.5/14)$ = 0.73 |
| 9.0 | $1 – \Phi(-9.0/14)$ = 0.74 |
6. Interpretation
- The model’s prediction for x = 3.5 (60%) closely matches the empirical estimate (61%).
- For large spreads (8.5 or 9 points), the model produces smoother and more plausible probabilities than the raw data, avoiding distortion from small sample noise.
- This shows the value of parametric modeling: it balances empirical data with reasonable structural assumptions to produce stable probability estimates.
7. Key Insights
- Point spreads represent market-based medians of belief about score differences.
- Empirical probabilities (from observed frequencies) are intuitive but unstable when data are sparse.
- A parametric model—assuming $d = y – x \sim N(0, 14^2)$—provides a good fit and smoother estimates.
- The model implies:
- The chance of covering the spread = 0.5
- The chance of winning outright = $1 – \Phi(-x/14)$
- This example demonstrates how probability models can systematically quantify uncertainty using both data and reasonable assumptions, even outside a full Bayesian framework.