Chapter 3 Key Takeaways: Expected Value and the Bettor's Edge

Summary Card -- Print or bookmark this page as a quick reference for the core formulas, concepts, and decision frameworks from Chapter 3.

Core Formulas

1. Expected Value (EV)

The expected value of a bet is the average profit or loss you would realize if you placed the same bet an infinite number of times.

$$\boxed{EV = (p \times W) - ((1 - p) \times L)}$$

Where: - $p$ = true probability of winning the bet (between 0 and 1) - $W$ = net profit if the bet wins - $L$ = amount lost if the bet loses (equal to the stake)

Using decimal odds ($d$) and stake ($S$):

$$EV = p \times S \times (d - 1) - (1 - p) \times S$$

Simplified:

$$EV = S \times \left[ p \times d - 1 \right]$$

2. Edge

Edge measures the difference between the bettor's estimated true probability and the probability implied by the sportsbook's odds.

$$\boxed{\text{Edge} = p_{\text{true}} - p_{\text{implied}}}$$

Where: - $p_{\text{true}}$ = the bettor's estimated true probability of the outcome occurring - $p_{\text{implied}}$ = the probability implied by the sportsbook's odds

Implied probability from American odds:

$$p_{\text{implied}} = \begin{cases} \frac{|\text{odds}|}{|\text{odds}| + 100} & \text{if odds are negative (favorite)} \\ \frac{100}{\text{odds} + 100} & \text{if odds are positive (underdog)} \end{cases}$$

Implied probability from decimal odds:

$$p_{\text{implied}} = \frac{1}{d}$$

3. Yield (Return on Investment)

Yield measures the bettor's actual profitability as a percentage of total money wagered.

$$\boxed{\text{Yield} = \frac{\text{Total Profit (or Loss)}}{\text{Total Amount Staked}} \times 100\%}$$

Reference benchmarks:

Important: Yield is only meaningful with a sufficiently large sample size. A +10% yield over 50 bets is far less informative than a +2% yield over 2,000 bets.

4. Break-Even Win Rate

The win rate required for $EV = 0$ at given odds.

$$\boxed{p_{\text{break-even}} = \frac{1}{d}}$$

Where $d$ is the decimal odds. Equivalently, for American odds:

$$p_{\text{break-even}} = \begin{cases} \frac{|\text{odds}|}{|\text{odds}| + 100} & \text{if negative odds} \\ \frac{100}{\text{odds} + 100} & \text{if positive odds} \end{cases}$$

Common break-even rates:

5. Vigorish (Vig) / Overround

The sportsbook's built-in margin, calculated from the total implied probabilities.

$$\boxed{\text{Overround} = p_{\text{implied, A}} + p_{\text{implied, B}}}$$

$$\boxed{\text{Vig \%} = \left(1 - \frac{1}{\text{Overround}}\right) \times 100\%}$$

Example: Both sides at -110 odds.

$$\text{Overround} = 0.5238 + 0.5238 = 1.0476$$ $$\text{Vig \%} = \left(1 - \frac{1}{1.0476}\right) \times 100\% = 4.55\%$$

6. Variance of a Single Bet

$$\boxed{\text{Var}(X) = p \times (W - \mu)^2 + (1 - p) \times (-L - \mu)^2}$$

Where $\mu = EV$ per bet, $W$ = profit if win, $L$ = stake (loss if lose).

Standard deviation of total profit over $n$ independent bets:

$$\sigma_{\text{total}} = \sigma_{\text{per bet}} \times \sqrt{n}$$

95% Confidence interval for total profit:

$$\text{CI}_{95\%} = n \times \mu \pm 1.96 \times \sigma_{\text{per bet}} \times \sqrt{n}$$

Key Concepts

The Central Principle

The only sustainable path to profitability in sports betting is to consistently identify and bet on positive expected value (+EV) opportunities.

No money management system, no staking strategy, and no amount of intuition can overcome consistently negative EV bets. Expected value is the single number that determines whether a bet is worth making.

The Law of Large Numbers (LLN)

The LLN guarantees that over a sufficiently large number of independent bets, the bettor's average result per bet will converge to the expected value.

What this means for bettors: - A +EV bettor will be profitable in the long run with near certainty. - A -EV bettor will lose money in the long run with near certainty. - Short-term results (dozens or even hundreds of bets) can deviate dramatically from EV due to variance. - The sportsbook leverages the LLN by processing millions of bets, ensuring its profit margin is realized with very high precision.

What this does NOT mean: - It does NOT mean that losses will be "corrected" by future wins (gambler's fallacy). - It does NOT guarantee any specific result over any finite number of bets. - It does NOT eliminate the need for bankroll management.

Variance: The Short-Run Enemy of the +EV Bettor

Even with a genuine mathematical edge, short-term results are dominated by randomness. Key variance principles:

1. Higher odds = higher variance. A +300 underdog bettor experiences far more volatility than a -110 spread bettor, even with the same edge.

2. Longer losing streaks are normal. A bettor with a 55% win rate at -110 odds will experience losing streaks of 8+ bets with regularity over a season.

3. Sample size matters enormously. It typically takes 1,000-2,500 bets to statistically confirm a 2-3% edge with 95% confidence.

4. Bankroll management is the bridge. Proper bet sizing (typically 1-3% of bankroll per bet) ensures survival through inevitable downswings.

Closing Line Value (CLV)

Closing line value measures whether a bettor consistently obtains odds that are better than the final closing line.

• Positive CLV = bettor got better odds than the market's final assessment = strong evidence of +EV betting.
• CLV is a more reliable indicator of skill than win-loss record because it is measurable on every bet and has lower variance.
• Sportsbooks use CLV to identify and restrict sharp bettors.

The Vig Creates a Default -EV Environment

Every bet at a sportsbook starts with a negative expected value for the bettor because the vig inflates the implied probabilities above 100%. The bettor must overcome this built-in disadvantage by estimating true probabilities more accurately than the market-implied probabilities.

At standard -110/-110 odds: - The vig is approximately 4.55%. - The bettor must win at least 52.38% of bets just to break even. - Every percentage point of win rate above 52.38% translates to approximately 1.9 cents of profit per dollar wagered.

Decision Framework: When to Bet

Use this structured checklist before placing any wager:

Step 1: Estimate the True Probability

• What is your best estimate of the true probability of the outcome?
• Is this estimate based on a rigorous model, data analysis, or strong qualitative evidence?
• How confident are you in this estimate?

Step 2: Calculate the Implied Probability

• What are the sportsbook's odds for this bet?
• Convert the odds to implied probability.

Step 3: Determine Your Edge

$$\text{Edge} = p_{\text{true}} - p_{\text{implied}}$$

• Is your edge positive?
• Is your edge large enough to be meaningful (typically at least 2-3% to overcome estimation error)?

Step 4: Calculate the Expected Value

$$EV = S \times [p_{\text{true}} \times d - 1]$$

• Is the EV positive?
• What is the EV as a percentage of the stake?

Step 5: Check Bankroll Constraints

• Does the optimal bet size (from Kelly or fractional Kelly) fit within your bankroll management rules?
• Is the bet size no more than 1-5% of your bankroll (depending on your risk tolerance and edge confidence)?
• Can you withstand a losing streak of 15-20 bets at this stake level without significant bankroll damage?

Step 6: Consider Practical Factors

• Are you getting the best available odds (line shopping)?
• Is the bet available at a sportsbook that will not restrict your account for winning?
• Is the bet size within the sportsbook's limits?
• Are you betting too many correlated outcomes (e.g., same game, same team)?

Step 7: Make the Decision

Quick Reference: EV Calculation Examples

Example 1: Favorite at -150

• Stake: $100
• True probability: 63%
• Win profit: $100 / 1.50 = $66.67
• Implied probability: 60.00%
• Edge: 63% - 60% = +3.00%
• EV: (0.63 x $66.67) - (0.37 x $100) = $42.00 - $37.00 = +$5.00

Example 2: Underdog at +200

• Stake: $100
• True probability: 36%
• Win profit: $200
• Implied probability: 33.33%
• Edge: 36% - 33.33% = +2.67%
• EV: (0.36 x $200) - (0.64 x $100) = $72.00 - $64.00 = +$8.00

Example 3: Standard Spread at -110

• Stake: $100
• True probability: 54%
• Win profit: $90.91
• Implied probability: 52.38%
• Edge: 54% - 52.38% = +1.62%
• EV: (0.54 x $90.91) - (0.46 x $100) = $49.09 - $46.00 = +$3.09

Example 4: Negative EV Trap at -110

• Stake: $100
• True probability: 51%
• Win profit: $90.91
• Implied probability: 52.38%
• Edge: 51% - 52.38% = -1.38%
• EV: (0.51 x $90.91) - (0.49 x $100) = $46.36 - $49.00 = -$2.64
• This bet feels like a winner (51% chance!) but is -EV due to the vig.

Common Pitfalls to Avoid

Chapter 3 in One Sentence

Expected value is the single number that determines whether a bet is worth making: calculate it, bet only when it is positive, size your bets to survive variance, and trust the Law of Large Numbers to deliver your edge over time.

Next Chapter: Chapter 4 covers Bankroll Management and Staking Strategies, where we explore how to optimally size bets given your edge, bankroll, and risk tolerance -- turning the +EV bets identified in this chapter into a sustainable, long-term profitable strategy.