Common Fallacies in Player Prop Analysis

The Gambler's Fallacy: Misunderstanding Independence

The gambler's fallacy is the mistaken belief that past independent events affect future probabilities. In prop betting, it manifests as:

"Player A has gone under his point total in 5 straight games. He's due to go over tonight!"

Why This Is Wrong

If each game is an independent event (reasonable assumption for many props), the probability of going over tonight is unchanged by past results. Formally:

The conditional probability equals the unconditional probability for independent events. Past results provide no predictive information about tonight.

The Mathematical Reality

Suppose a player has a true 50% probability of going over his line each game (a fair prop). What's the probability of 5 straight unders?

This is rare (happens 1 in 32 times), which makes it feel like "he's due" for an over. But this is an illusion. The 3.125% probability applied before the streak began. Now that the streak has occurred, we're in a new situation:

The coin (or player) has no memory. Each game is a fresh 50-50 proposition.

When Past Results DO Matter

Past results are informative when they update our estimate of the underlying probability. If a player we thought was 50% to go over has gone under 10 straight times, either:

1. We got unlucky (0.5^10 = 0.1% probability), or
2. Our 50% estimate was wrong, and the true probability is lower

Bayesian reasoning suggests we should update toward option 2. But this is different from the gambler's fallacy—we're not saying "he's due," we're saying "our probability estimate may have been incorrect."

Worked Example

Player A's points prop is 24.5. You initially estimated 55% probability of over based on season-long data (n=50 games, 28 overs). He's now gone under in 5 straight games.

Gambler's fallacy response: "He's due! I'll bet big on the over!"

Correct Bayesian response: "This 5-game sample suggests my 55% estimate might be too high. With 50 total games and 28 overs (23 over, 27 under after these 5), my updated estimate is 23/50 = 46%. I should not bet the over."

The fallacy says past results make the opposite more likely. The correct view says past results help us estimate the true underlying probability.

The Hot Hand Fallacy vs. Real Streakiness

The hot hand fallacy is the opposite error: believing that recent success predicts future success beyond what the data warrant.

"Player B has gone over in his last 6 games. He's on fire! Bet the over!"

The Research

Classic psychology research (Gilovich, Vallone, & Tversky, 1985) analyzed basketball shooting and found no evidence that making several shots in a row increased the probability of making the next shot. Players were no more likely to make a shot after making several in a row than after missing several.

This suggests the "hot hand" is largely an illusion—humans see patterns in random sequences.

But Wait—Is Streakiness Real?

More recent research (Miller & Sanjurjo, 2018) showed the original analysis had a subtle statistical flaw. When properly analyzed, there is weak evidence of hot hand effects in basketball shooting (~2-4 percentage point increase).

So the truth is nuanced:

• Most perceived "hot hands" are random variance appearing as patterns
• True hot hand effects exist but are small (2-4 percentage points, not 20)
• Overweighting recent performance is still a fallacy even if mild streakiness is real

The Mathematical Test

How can you tell if a streak is real or random? Calculate the probability of observing the streak by chance.

Player with 50% true probability has 6 straight overs. Probability:

This is unlikely but not extraordinarily so. If 100 players each play 40 games, we'd expect several to have 6-game streaks by pure chance.

Proper interpretation: The streak is weak evidence the true probability exceeds 50%, but not strong evidence. We should modestly update our estimate (perhaps from 50% to 52-54%), not wildly revise it to 75%.

Regression to the Mean (Preview)

The hot hand fallacy fails to account for regression to the mean: extreme performances tend to be followed by less extreme performances. We'll explore this mathematically in the next section.

Regression to the Mean: The Mathematics

Regression to the mean is a statistical phenomenon, not a psychological bias. It's a mathematical necessity that extreme observations tend to be followed by less extreme observations.

Why It Occurs

Any observed performance has two components:

When we observe an extreme performance (very high or very low), it's likely that:

1. True skill is somewhat extreme, AND
2. Random variation was extreme in the same direction

In the next performance, we expect:

• True skill to remain the same
• Random variation to be closer to average (by definition of random)

Therefore, the next performance will likely be less extreme than the first—this is regression to the mean.

The Regression Formula

If a player's recent average is X_recent and their long-term average is X_longterm, the expected next performance is:

Where w is the weight given to recent data, which depends on:

• Sample size of recent data (larger sample → higher w)
• Consistency of player (more consistent → higher w)
• Reason for change (injury recovery → higher w; random hot streak → lower w)

A rough guideline for weight:

Where n_recent is the size of the recent sample and k is a constant (~30-50 for most player props) representing how much we trust long-term data.

Worked Example

Player C averages 6.2 rebounds per game over 200 career games. In his last 10 games, he's averaged 9.5 rebounds per game. What do we predict for tonight?

Naive approach: "He's averaging 9.5 recently, so predict 9.5."

Proper regression approach:

We predict 6.86 rebounds, much closer to his career average than his recent hot streak. This accounts for the strong possibility that his recent 9.5 average included positive random variance.

How Much Regression?

The amount of regression depends on sample sizes:

With only 5-10 games of "hot" performance, we should weight career data 80-90%. Most bettors do the opposite, weighting recent data far too heavily.

Sample Size Neglect: The Law of Small Numbers

Sample size neglect is the failure to account for how sample size affects confidence. Small samples have huge uncertainty, but bettors often treat them as reliable.

The Mathematical Reality

From Article 2, recall that standard error depends on sample size:

The 95% confidence interval width is approximately ±2 SE:

Critical insight: With 10 games showing 7 overs (70%), the 95% CI is [39%, 100%]. This is consistent with true probability anywhere from 39% to 100%. The data tell us almost nothing!

Worked Example

Two players:

Player E: 70% over rate in last 10 games (7-3)

Player F: 70% over rate in last 100 games (70-30)

Question: Which 70% should we trust more?

Player E (n=10):

Player F (n=100):

Player F's 70% is much more reliable. Player E's could easily be a 50% player who got lucky.

Minimum Sample Size Rule

For any split or subset analysis:

• n < 10: Ignore completely, pure noise
• n = 10-30: Weak evidence, use cautiously
• n = 30-50: Moderate evidence, worth considering
• n > 50: Strong evidence, reliable for estimation

Most prop bettors violate this constantly, trusting 5-10 game samples.

The Narrative Fallacy: Stories Over Statistics

The narrative fallacy is the tendency to construct explanatory stories around random or statistical phenomena, then use those stories to predict the future.

Common Narratives

• "He's motivated because he's playing his former team!"
• "They always play up to the competition!"
• "This is a must-win game, he'll step up!"
• "He's in a contract year, he'll be extra focused!"

Why Narratives Are Dangerous

These narratives might be true sometimes, but they suffer from:

1. Unfalsifiability: If he performs well, the narrative is confirmed. If he doesn't, we explain it away ("He was too motivated and pressed").
2. Hindsight bias: After the fact, we construct narratives that "explain" outcomes. This doesn't mean the narrative had predictive power.
3. Sample size=1: We remember the one time someone "played up to competition," not the 20 times they didn't.

The Test

Before betting based on a narrative, ask:

1. Is this testable? Can I gather data on past instances?
2. What does the data show? Do players actually perform better against former teams (on average, with adequate sample size)?
3. Is the effect priced in? If it's a known phenomenon, the bookmaker already adjusted for it.

Example: "Revenge Game" Narrative

Narrative: "Player G always goes off against his former team!"

Test it: Player G has played his former team 4 times since being traded. Outcomes: 28 pts, 18 pts, 32 pts, 22 pts. Average: 25 pts.

Career average: 24 pts (n=200 games).

Analysis:

The "revenge game" effect is not statistically distinguishable from zero. The narrative is unsupported by data.

When Narratives Matter

Narratives are useful when they point to structural changes you can verify with data:

• "He's healthy now after missing 20 games" → Check minutes, usage rate
• "New coach runs more plays for him" → Check shot attempts, touches per game
• "Team is tanking, he'll get more minutes" → Check actual minutes trend

But use the narrative to identify what to test, not as the test itself.

Summary: Fallacies and Their Antidotes

Building a Fallacy-Resistant Betting Process

To systematically avoid these fallacies, build a consistent analytical process:

1. Use a Standardized Framework

Follow the same steps for every prop, building on the complete framework from Articles 1-4:

1. Extract market information using techniques from Article 1 (odds conversion, hold calculation)
2. Gather historical data and calculate confidence intervals (Article 2)
3. Weight by sample size and apply regression to the mean (Article 2)
4. Apply contextual adjustments conservatively (Article 2)
5. Calculate expected value (Article 2)
6. Size bet using Kelly Criterion (Article 3)
7. Account for correlation if betting multiple props from same game (Article 4)

Never deviate from your process based on "feelings" or "gut instinct."

2. Keep a Decision Journal

For each bet, record:

• Your probability estimate and reasoning
• What data you considered
• What data you ignored and why
• Result and actual player performance

Review quarterly: Are you falling into patterns? Overweighting recent games? Cherry-picking stats?

3. Calculate Calibration

After 50+ bets, test your calibration:

• When you estimate 55% probability, do props hit ~55% of the time?
• When you estimate 65% probability, do props hit ~65% of the time?

If you're poorly calibrated (estimating 60% but hitting 50%), you're falling into fallacies—likely overconfidence and confirmation bias.

4. Use Base Rates

Always start with the base rate (career average, season average) and require strong evidence to deviate. The burden of proof is on recent data to overcome the large-sample career data.

5. Embrace Uncertainty

Express estimates as ranges, not point values:

• Bad: "I estimate exactly 57.3% probability"
• Good: "I estimate 54-60% probability with best guess 57%"

This humility prevents overconfidence and excessive bet sizing.

6. Seek Disconfirming Evidence

Before making a bet, deliberately search for reasons NOT to bet it. If you can't find any contradicting evidence, you're not looking hard enough—confirmation bias is at work.