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Expected Value in Player Prop Betting
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The Mathematics of Player Props - Article 2 of 5
Series Navigation:
- Article 1: Understanding the Math Behind the Lines
- Article 2: Expected Value in Player Prop Betting (You are here)
- Article 3: Variance and Bankroll Management for Props
- Article 4: Same-Game Parlays: The Mathematics of Correlation
- Article 5: Common Fallacies in Player Prop Analysis
The Mathematics of Identifying Positive Expected Value Opportunities
Introduction
Disclaimer: This article is for educational purposes only and is not betting advice. The examples used are hypothetical and illustrative. I cannot predict outcomes or guarantee profits. The goal is to teach the mathematical framework for evaluating expected value.
In Article 1, we learned how to extract information from betting lines: converting odds to probability, calculating the bookmaker's hold, and identifying fair probabilities. This taught us what the market is saying.
But knowing what the market says isn't enough. To make informed betting decisions, you need to answer a more fundamental question: Is this bet worth making?
This question is answered through expected value (EV) analysis. Expected value is the mathematically rigorous way to evaluate any bet, investment, or decision under uncertainty. It tells you, on average over many repetitions, how much you stand to gain or lose per dollar wagered.
In this article, we'll cover:
- The mathematical definition and calculation of expected value
- How to estimate true probability from historical data
- Sample size requirements and confidence intervals
- A complete framework for evaluating player props
- Common mistakes in EV estimation
By the end, you'll understand how to determine whether a prop bet offers positive expected value—and more importantly, you'll understand the limitations and uncertainties inherent in that determination.
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Expected Value: Mathematical Definition
Expected value is the weighted average of all possible outcomes, where each outcome is weighted by its probability of occurrence.
The General Formula
For any bet with multiple possible outcomes:
Where the sum is taken over all possible outcomes i.
The Binary Bet Formula (Win or Lose)
Most player props are binary: you either win or lose. For these bets, the formula simplifies to:
Since we typically wager $1 (or express everything per dollar wagered), and we lose our entire wager if we lose, this becomes:
And since P_lose = 1 - P_win, we can write:
Understanding the Components
- P_win: The true probability of winning the bet (your estimate, not the market's)
- Profit: How much you profit per dollar wagered if you win (in decimal odds terms: decimal_odds - 1)
- EV > 0: Positive expected value—you expect to profit over many repetitions
- EV = 0: Break-even bet—no edge either way
- EV < 0: Negative expected value—you expect to lose over many repetitions
Worked Example 1: Simple EV Calculation
Let's calculate expected value for a hypothetical prop bet.
The Bet
Player A: Over 25.5 points at -110 odds
Step 1: Convert Odds to Decimal
From Article 1, we know -110 converts to:
This means if you bet $1 and win, you get back $1.909 (your $1 stake plus $0.909 profit).
Step 2: Determine Profit Per Dollar
So you profit $0.909 per $1 wagered if you win.
Step 3: Estimate True Probability
This is the crucial step. Suppose you've analyzed Player A's performance and determined (through methods we'll discuss shortly) that he has a 55% true probability of scoring over 25.5 points.
Step 4: Calculate Expected Value
Interpretation
With these assumptions, this bet has an expected value of +$0.05 per dollar wagered, or a 5% return on investment. Over 100 such bets, you would expect to profit $5 per $1 wagered on each (total profit: $5 on $100 wagered).
Critical caveat: This calculation is only as good as your probability estimate. If your true probability estimate is wrong, the EV calculation is wrong. We'll discuss how to form probability estimates next.
Your Estimate vs. The Market
For a bet to have positive expected value, your probability estimate must differ from (and be more accurate than) the market's probability in the right direction.
The Relationship
From Article 1, we learned that -110 odds imply a 52.4% probability (including vig). The fair probability (removing vig) is approximately 50% in a balanced two-way market.
For our example:
- Market implied probability (with vig): 52.4%
- Market fair probability (estimated): ~50%
- Your estimate: 55%
You believe the true probability is 55%, while the market (after removing vig) believes it's closer to 50%. This 5 percentage point edge is what creates positive expected value.
The Break-Even Probability
At what true probability does a bet at -110 become break-even (EV = 0)?
You need a 52.4% true win probability just to break even at -110 odds. This is exactly the implied probability we calculated in Article 1. To have positive EV, your estimated true probability must exceed 52.4%.
General Break-Even Formula
For any odds, the break-even probability is:
This is mathematically equivalent to the implied probability from Article 1.
Estimating True Probability from Historical Data
The most fundamental challenge in EV analysis: How do you estimate true probability?
The most common approach is to use historical data on the player's past performance.
The Naive Approach
Suppose Player A has played 50 games this season, and in 28 of those games, he scored over 25.5 points.
Is 56% our probability estimate? Not so fast. This sample proportion is our starting point, but we need to account for:
- Sample size uncertainty
- Context differences (opponent strength, home/away, rest days, etc.)
- Recent trends vs. season-long averages
- Regression to the mean
Sample Size and Standard Error
With only 50 games, there is considerable uncertainty around the 56% estimate. We quantify this using the standard error of a proportion:
Where p is the sample proportion and n is the sample size.
For our example:
The standard error is 7 percentage points. This is substantial uncertainty!
Confidence Intervals
A 95% confidence interval is approximately:
Interpretation: With 95% confidence, the true probability lies somewhere between 42.3% and 69.7%. This is a massive range! At the low end (42.3%), the bet would have terrible EV. At the high end (69.7%), it would have enormous +EV.
This uncertainty is why sample size matters so much in probability estimation.
How Much Data Do You Need?
The table below shows the standard error and 95% confidence interval width for different sample sizes, assuming p = 0.50 (maximum uncertainty occurs at p = 0.50).
| Sample Size (n) | Standard Error | 95% CI Width (±) |
|---|---|---|
| 10 games | 15.8% | ±31.0% |
| 25 games | 10.0% | ±19.6% |
| 50 games | 7.1% | ±13.9% |
| 100 games | 5.0% | ±9.8% |
| 200 games | 3.5% | ±6.9% |
| 500 games | 2.2% | ±4.4% |
| 1000 games | 1.6% | ±3.1% |
Key Observations
- With 10 games: 95% confidence interval is ±31%, making probability estimates nearly useless
- With 50 games: 95% confidence interval is ±14%, still substantial uncertainty
- With 200 games: 95% confidence interval is ±7%, reasonable for estimation
- With 1000 games: 95% confidence interval is ±3%, good precision
The problem: Most players don't have 200+ game samples in a single context. An NBA season is only 82 games. An NFL season is only 17 games. This limited data creates fundamental uncertainty in probability estimation.
Worked Example 2: EV with Proper Uncertainty Analysis
Let's work through a more realistic example that accounts for sample size limitations.
The Bet
Player B: Over 6.5 rebounds at +105 odds
Step 1: Historical Data
Player B has played 40 games this season. In 22 of those games, he recorded over 6.5 rebounds.
Standard error = √[0.55 × 0.45 / 40] = 0.079 = 7.9%
95% CI = 0.55 ± (1.96 × 0.079) = 0.55 ± 0.155 = [0.395, 0.705]
Step 2: Contextual Adjustments
You notice that in his last 10 games (smaller opponent frontcourts), he's gone over 6.5 rebounds in 8 games (80%). However, tonight's opponent has a strong rebounding team. Looking at games against top-10 rebounding teams, he's only 3-for-8 (37.5%).
Which estimate should you use?
- Season-long: 55% (n=40)
- Last 10 games: 80% (n=10, but small sample!)
- vs. top rebounding teams: 37.5% (n=8, very small sample!)
The season-long 55% has the most data, but may not reflect tonight's matchup. The top-rebounding-team sample (37.5%) is most relevant but has very high uncertainty (SE ≈ 17%).
A reasonable approach: Weight the general estimate more heavily due to larger sample size, with modest adjustment for context. Let's estimate 48% as our true probability (between the 55% season average and 37.5% tough-matchup rate).
Step 3: Calculate EV
Odds: +105 → Decimal odds: 2.05 → Profit per dollar: 1.05
Step 4: Sensitivity Analysis
Given our uncertainty, let's calculate EV at the confidence interval boundaries:
Optimistic case (58% true probability):
Pessimistic case (38% true probability):
Conclusion
Our best estimate suggests this is a marginally -EV bet (-1.6%). However, given our uncertainty (confidence interval includes both strongly +EV and strongly -EV scenarios), this is not a clear-cut decision. A conservative bettor would pass. An aggressive bettor who believes their contextual adjustment is accurate might still bet if they think the true probability is closer to 50%+.
Key lesson: Uncertainty is part of the analysis. Don't pretend you know true probability with precision you don't have.
The Regression to the Mean Problem
One of the most common errors in prop betting is failing to account for regression to the mean.
What Is Regression to the Mean?
When a player performs unusually well (or poorly) over a small sample, we expect their future performance to "regress" back toward their long-term average. This is a mathematical necessity, not a psychological phenomenon.
Example
Player C averages 18 points per game over his career (500 games). In his last 10 games, he's averaged 26 points per game.
Naive analysis: "He's averaging 26 PPG recently, so he'll probably score over 24.5 tonight!"
Statistical reality: The 26 PPG is likely inflated by random variance. We expect his next game to be somewhere between 18 (career average) and 26 (recent hot streak), weighted by the strength of each sample.
The Regression Formula
A simplified regression-to-mean formula:
Where weights w₁ and w₂ depend on sample sizes. Larger samples get more weight.
For our example, if we weight the 500-game career sample much more heavily than the 10-game hot streak:
Our prediction is 19.2 PPG, much closer to his career average than his recent hot streak. This dramatically affects whether we'd bet over 24.5 points.
Bottom line: Hot and cold streaks mean less than you think. Larger sample sizes (career averages, season-long data) deserve significant weight even when recent performance differs. We'll explore this more in Article 5 on common fallacies, particularly the hot hand fallacy and proper regression to the mean.
A Complete Framework for Prop Evaluation
Bringing everything together, here's a step-by-step framework for evaluating any player prop:
Step 1: Extract Market Information
- Convert odds to implied probability
- Calculate bookmaker's hold
- Estimate fair probability (remove vig)
- Identify break-even probability
Step 2: Gather Historical Data
- Season-long performance: How often has player exceeded this line?
- Sample size: How many games? (More is better)
- Calculate sample proportion and standard error
- Construct confidence interval around estimate
Step 3: Make Contextual Adjustments
- Opponent strength in relevant category
- Home vs. away splits (if sample size sufficient)
- Rest days and back-to-backs
- Injury status (player and teammates)
- Recent trends (but weight carefully due to small samples)
Step 4: Form Probability Estimate
- Weight season-long data heavily (large sample)
- Adjust moderately for strong contextual factors
- Account for regression to mean on streaks
- Be conservative: If uncertain, shade toward market probability
Step 5: Calculate Expected Value
- Use formula: EV = (P_win × Profit) - (P_lose × 1)
- Perform sensitivity analysis: What if probability is ±5%?
- Consider confidence interval: Range of possible EVs
Step 6: Make Decision
- Only bet if EV is clearly positive (e.g., +3% or better)
- Pass on marginal situations where uncertainty is high
- Never bet just because you "have a feeling"
- Track your bets and review results to calibrate your estimates
Critical note: This framework doesn't guarantee profit. It's a systematic approach to thinking probabilistically. Even with perfect methodology, variance will create wins and losses. The goal is positive EV over many bets, not winning every individual bet.
Common Mistakes in EV Estimation
1. Over-Weighting Small Samples
"He's gone over in 4 of his last 5 games, so he'll probably go over tonight!"
Problem: 5 games is far too small a sample. The standard error is ~22%, making the estimate nearly worthless. A 4-1 record could easily be a 50% player who got lucky.
2. Ignoring Regression to Mean
"He's shooting 50% from three over his last 10 games vs. 35% career. He's clearly improved!"
Problem: Small samples create apparent trends that are just noise. Unless there's a mechanical reason for improvement (injury recovery, coaching change), assume regression toward career average.
3. False Precision
"I estimate exactly 53.7% probability based on my model."
Problem: With limited data, claiming precision to the nearest 0.1% is nonsense. Your uncertainty is likely ±5-10%. Acknowledge this in your analysis.
4. Confirmation Bias
"I really like this over, let me find stats that support it."
Problem: You can always find cherry-picked stats to support any position. Use a systematic framework and follow it consistently, even when it contradicts your intuition.
5. Ignoring Correlation
"I bet five different props from the same game, all independently +EV!"
Problem: Props from the same game are correlated. If the game goes differently than expected (blowout, low-scoring, etc.), multiple props may lose together. This creates portfolio risk we'll discuss in Article 3, and the mathematics of correlation are covered thoroughly in Article 4.
6. Not Tracking Results
"I think I'm profitable, but I don't keep records."
Problem: Without data, you can't improve your probability estimates or know if your approach is working. Track every bet: date, prop, odds, your estimated probability, result, and profit/loss.
Why Most Props Are -EV (And That's OK)
Here's an uncomfortable truth: Most player props offered by sportsbooks are -EV for the bettor. This isn't surprising—it's by design.
The Math Explains Why
From Article 1, we know typical holds are 4-6% for major props, higher for exotic props. This means:
Fair odds: Market prices at ~50% (after removing vig)
Your edge needed: You must estimate >52.4% true probability to have +EV
For a bet at -110 to be +EV, your probability estimate must be at least 2.4 percentage points higher than the market's fair estimate. Given the uncertainty in small-sample estimation (typically ±5-10%), finding clear +EV situations is rare.
The Sharps vs. The Market
Professional bettors ("sharps") spend enormous resources on data, models, and information gathering. The market's closing line represents the collective wisdom of these sharp bettors plus the bookmaker's models. Consistently beating this consensus is extremely difficult.
Where Value Might Exist
If +EV opportunities exist, they're most likely in:
- Secondary players: Less sharp attention, less modeling sophistication
- Niche stats: Exotic props where the bookmaker has less data
- Live betting: Rapidly changing situations where lines lag reality
- Late-breaking information: Injury news, lineup changes not yet priced in
But even in these markets, the bookmaker's hold is typically higher (10-15%+), requiring larger edges to overcome.
The Takeaway
Don't expect to find +EV props easily. If you're finding +EV bets everywhere, you're probably over-estimating your edge. Be skeptical of your own estimates, especially when they differ significantly from the market.
Practical Advice for EV-Based Prop Betting
1. Focus on Your Edge
Don't bet just because you have an opinion. Bet only when you have a genuine informational or analytical edge over the market. If you're using the same data everyone else has, you probably don't have an edge.
2. Bet Small When Uncertain
When your probability estimate has wide confidence intervals, bet smaller (or don't bet at all). Save your biggest bets for situations where you have high confidence in your estimate. We'll formalize this in Article 3 with the Kelly Criterion, which mathematically determines optimal bet sizing.
3. Specialize
Instead of betting many sports/leagues, focus on one or two where you can develop genuine expertise. Watch every game, track contextual factors, build statistical models. Specialization is how you develop edges the market doesn't have.
4. Track Everything
Keep detailed records:
- Date and prop description
- Odds and your estimated probability
- Your reasoning for the probability estimate
- Result (win/loss) and profit/loss
- Actual player performance
After 100+ bets, analyze: Are your probability estimates well-calibrated? When you estimate 55%, do props hit ~55% of the time? If not, adjust your methodology.
5. Accept Variance
Even with perfect +EV betting, you'll have losing streaks due to variance. A 60% win rate (excellent!) still means 40 losses per 100 bets. Don't abandon a sound approach after a bad run. We'll explore variance deeply in Article 3.
6. Know When to Walk Away
If after 100+ carefully tracked bets you're showing consistent losses, one of two things is true:
- You've been unlucky (possible but unlikely over 100+ bets)
- You don't actually have an edge (more likely)
Be honest with yourself. Most bettors don't have a genuine edge. That's not a moral failing—it's just very difficult to beat efficient markets.
Conclusion
Expected value analysis is the fundamental tool for evaluating any bet under uncertainty. The key concepts we've covered:
- EV formula: EV = (P_win × Profit) - (P_lose × 1). A bet is only worth making if EV > 0 with sufficient confidence.
- True probability estimation: Use historical data as a starting point, but account for sample size limitations through standard error and confidence intervals.
- Sample size matters enormously: With 50 games, your uncertainty is ±14%. With 200 games, it's ±7%. Most probability estimates have far more uncertainty than bettors acknowledge.
- Regression to the mean: Hot streaks and cold streaks are often noise. Weight long-term data more heavily than recent performance, especially with small recent samples.
- Systematic framework: Follow a consistent process for every prop: extract market info, gather data, make adjustments, estimate probability, calculate EV, make decision.
- Most props are -EV: This is by design. The bookmaker's hold ensures that betting randomly loses money. Finding true +EV requires genuine edge over the market.
What we haven't addressed: How should you size your bets? Even if you've correctly identified a +EV opportunity, betting too much (or too little) can be costly. Optimal bet sizing requires understanding variance and risk of ruin.
In Article 3: Variance and Bankroll Management for Props, we'll explore the mathematics of bet sizing, apply the Kelly Criterion to player props, calculate risk of ruin, and develop a portfolio approach for multiple prop bets. Expected value tells you what to bet; bankroll management tells you how much to bet.
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Series Navigation
The Mathematics of Player Props - Article 2 of 5
- Article 1: Understanding the Math Behind the Lines
- Article 2: Expected Value in Player Prop Betting (Current article)
- Article 3: Variance and Bankroll Management for Props
- Article 4: Same-Game Parlays: The Mathematics of Correlation
- Article 5: Common Fallacies in Player Prop Analysis
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- Expected Value in Gambling - General EV concepts across all gambling
- Why Betting Systems Fail - Understanding negative expectation
- Sports Betting Mathematics - Broader sports betting analysis