Example — Probabilities from Football Point Spreads#

Part 1 · Stage 1 · 🎲 The Bayesian Idea · Lesson 006 of 144 · beginner

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Assignment, not inference#

This example illustrates probability assignment — how to arrive at a number — rather than Bayesian inference itself. Its subject is the American-football point spread: the bookmakers’ published prediction of the margin by which the favourite will win. Given a spread, what is the probability the favourite actually covers it, or simply wins?

Three routes to a number#

The same question is approached three ways, matching the three justifications of the previous lesson:

Subjective — an informed fan states a probability directly.
Empirical — count outcomes in a database of games. Across 672 professional games, one can simply tabulate how often favourites at a given spread won.
Parametric — build a probability model for the outcome and read the probability off it.

The parametric model#

The empirical route runs out of data at any particular spread, so the model earns its keep. Plotting \(d = (\text{actual outcome}) - (\text{point spread})\) against the spread shows the differences are roughly centred at zero with a spread of about 14 points, and largely independent of the spread itself. That suggests

\[d \sim \mathrm{N}(0,\; 14^2),\]

so the favourite (spread \(s\)) wins when the actual margin exceeds 0, i.e. when \(d > -s\):

from scipy.stats import norm
s = 3.5                                  # point spread
p_win = 1 - norm.cdf(-s, loc=0, scale=14)   # P(favourite wins)  ≈ 0.60
p_cover = 1 - norm.cdf(0, loc=0, scale=14)  # P(covers spread)   = 0.50

The lessons#

Two. First, the model smooths and extrapolates: it gives a probability at spreads where few games were ever played, which raw counts cannot. Second, the model is checked against data — the zero-centred, constant-variance normal is adopted because the scatterplot supports it, not because it is convenient. Probability assignment, done honestly, already involves the third of the three steps.

Hint