Bayes, Base Rates, and Bad Beats: Avoiding a Common Pitfall in Probability

Ronald Lockington
Oct 29, 2024
3 min read

Updated: 23 hours ago

Summary: If you ignore base rates, you’ll misread injury news, hot streaks, and market moves. Bayesian reasoning fixes that by combining the big picture (priors) with new info (evidence) to get a smarter, updated probability.

What is the Base Rate Fallacy?

The base rate fallacy is the mistake of ignoring how common something is (its base rate) and focusing only on vivid details.

Classic example: You meet a shy, thin, detail-oriented man with glasses. Is he more likely a construction worker or a marine biologist?

Most people guess “marine biologist” because the description fits, but in the U.S. there are far more construction workers than marine biologists.

Base rates matter: a random person is much more likely to be a construction worker, even if the description feels like a match.

Sports example: A rookie starts hot over a few games. Treating those few results as the “truth” ignores the base rate that most rookies regress toward league averages.

Smarter: compare to a cohort of similar rookies (position, age, draft capital, scouting grades) and update as you learn more.

The Fix: Bayesian Reasoning (Plain English)

Bayesian reasoning blends what you already know (prior) with new evidence to produce a revised belief (posterior).

The math looks intimidating, but isn't too bad:

Bayes' Theorem: P(A|B) = [P(A) * P(B|A)] / P(B)

What each of these mean:

P(A|B) = Probability of A conditional on B = Posterior Probability.

P(A) = Probability of A = Prior probability.

P(B|A) = Probability of B conditional on A. P(B) = Probability of B.

And in plain English:

Updated belief = (Prior belief * How well the new info fits the prior belief) ÷ (How common that info is overall).

A Basketball Injury Example (Step-by-Step)

Question: What’s the chance an injured player plays, if you know he had a full practice the day before a game?

Prior, P(Play): 0.20 (20%) based on injury reports
Likelihood, P(Practice ∣ Play): 0.90 (most players who suit up fully practice the day before)
Evidence rate, P(Practice): 0.25 (regardless of game status, 25% chance he’d fully practice)

Plug it in:

P(Play ∣ Practice)= 0.20× 0.90 / .25 = 0.72

Posterior: 72% chance he plays.

Why this helps: You don’t jump to “he practiced, so he’s in.” You weight the practice by how frequently full practices happen overall.

Revisiting the Occupation Example (With Numbers)

P(Marine Biologist): about 1% (≈13k male marine biologists vs ≈1.3m male construction workers)
P(Characteristics ∣ Marine Biologist): say 75%
P(Characteristics): say 10% of men fit that description in general

P(Marine Biologist ∣ Characteristics) = 0.01 × 0.750 / .10 = 0.075 = 7.5%

Even with a strong “fit,” the low base rate keeps the posterior under 10%.

Practical Betting Uses (U.S. Books & Markets)

Injury news & line moves: Update from a prior (historical absence rates by injury type) using fresh reports (limited practice, full practice, beat-writer videos).
Hot streaks / slumps: Start from league/base rates, then adjust with new splits or role changes (usage, minutes, matchup).
Live betting: Combine pre-game priors (team strength, totals) with in-game evidence (pace, shot quality, foul trouble) to keep your probability curve calibrated.
Props & derivatives: Anchor to no-vig fair odds (market priors), then adjust for player-specific news before you compare prices across FanDuel, DraftKings, BetMGM, etc.

Rule of thumb: The rarer the evidence, the more it moves your belief. The more common the evidence, the less weight it deserves.

Common Mistakes to Avoid

Over-reacting to common evidence (e.g., “He warmed up!” when most Questionable players warm up anyway).
Using the wrong base rate (league average when you need a position-, role-, or cohort-specific rate).
Double-counting the same evidence (e.g., practice report + beat-writer tweet about the same practice).
Ignoring vig: Always compare to no-vig benchmarks before updating your view of value.

TL;DR: A Three-Step Bayesian Habit

Start with a prior: league/cohort base rate or a no-vig market probability.
Score the evidence: estimate how likely that news would be if your prior were true vs in general.
Update, don’t overreact: adjust the probability; then check if the price you’re being offered beats your new fair line.