.. _bda-example-probabilities-from-football-point-spreads: ======================================================================== Example — Probabilities from Football Point Spreads ======================================================================== **Part 1 · Stage 1 · 🎲 The Bayesian Idea** · Lesson 006 of 144 · *beginner* :doc:`◀ Previous · Probability as a Measure of Uncertainty <005-probability-as-a-measure-of-uncertainty>` · :doc:`Next · Example — Calibration for Record Linkage ▶ <007-example-calibration-for-record-linkage>` · :doc:`↑ Section ` .. important:: **✨ AI-generated content.** This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it. 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 :math:`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 .. math:: d \sim \mathrm{N}(0,\; 14^2), so the favourite (spread :math:`s`) wins when the actual margin exceeds 0, i.e. when :math:`d > -s`: .. code-block:: python 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:: **Related lessons:** :doc:`Probability as a Measure of Uncertainty <005-probability-as-a-measure-of-uncertainty>` · :doc:`Example — Calibration for Record Linkage <007-example-calibration-for-record-linkage>` · :doc:`Some Useful Results from Probability Theory <008-some-useful-results-from-probability-theory>` · :doc:`Normal Distribution with Known Variance <015-normal-distribution-with-known-variance>` .. seealso:: **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/08/example-probabilities-from-football-point-spreads/ `__ (insightful-data-lab.com). .. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: beginner