Chapter 2 Key Takeaways: Probability and Odds
Core Concepts
Probability is the language of sports betting. Every line, every odds number, and every betting decision is ultimately a statement about probability. Understanding how to translate between odds formats and probability is the foundational skill upon which all sports betting analysis is built.
Odds are distorted probabilities. Sportsbook odds do not represent true probabilities. They are true probabilities adjusted by the bookmaker's margin (the vig or overround). The bettor's first task when evaluating any line is to strip away the vig to estimate what the book believes the true probability to be, and then compare that to their own estimate.
Value is the only thing that matters. A bet has value when the bettor's estimated true probability of an outcome exceeds the implied probability embedded in the odds. Consistently finding positive expected value (+EV) bets is the sole mechanism for long-term profitability. Betting without an edge is guaranteed to lose money over time because of the vig.
The Law of Large Numbers is both your ally and your tormentor. In the long run, results converge to expected value. But the "long run" in sports betting can require thousands of bets. Short-run variance can be brutal, and a positive edge does not protect against significant drawdowns over hundreds of wagers.
Independence matters. Evaluating whether events are truly independent is critical for parlay construction, bankroll management, and probability estimation. Most sporting events are not perfectly independent, and correlated outcomes require more sophisticated analysis.
Master Conversion Table
| From / To | American (Negative) | American (Positive) | Decimal | Fractional | Implied Probability |
|---|---|---|---|---|---|
| American (-) | -- | -- | 1 + 100/|A| | 100/|A| (as fraction) | |A| / (|A| + 100) |
| American (+) | -- | -- | 1 + A/100 | A/100 (as fraction) | 100 / (A + 100) |
| Decimal | -100/(D-1) [if D<2] | (D-1) x 100 [if D>=2] | -- | (D-1) as fraction | 1 / D |
| Fractional (N/D) | -100/(N/D) [if N<D] | (N/D) x 100 [if N>=D] | 1 + N/D | -- | D / (N + D) |
| Implied Prob (p) | -100p/(1-p) [if p>0.5] | 100(1-p)/p [if p<=0.5] | 1 / p | (1-p)/p as fraction | -- |
Where A = American odds value, D = decimal odds value, N/D = fractional numerator/denominator, p = probability.
Key Formulas at a Glance
| Formula | Expression |
|---|---|
| Overround | Sum of all implied probabilities - 100% |
| Margin on turnover | (Overround) / (1 + Overround) |
| Vig-removed probability (multiplicative) | Implied probability / Sum of implied probabilities |
| Parlay decimal odds | Product of individual decimal odds |
| Expected value | (P_win x Profit) - (P_lose x Stake) |
| Brier score | (1/N) x Sum of (predicted - actual)^2 |
| Arbitrage condition | Sum of (1 / best decimal odds per outcome) < 1 |
Decision Framework: Should I Bet?
Step 1 -- Estimate True Probability. Use your model, research, or analysis to arrive at an honest estimate of the true probability of the outcome.
Step 2 -- Calculate Implied Probability. Convert the sportsbook's odds into implied probability. This tells you what probability you are "paying for."
Step 3 -- Compare. If your estimated true probability is greater than the implied probability, the bet has positive expected value. If not, pass.
Step 4 -- Assess the Edge Size. A tiny edge (e.g., 0.5%) may not justify the variance risk. Consider the magnitude of the edge relative to the odds and your bankroll.
Step 5 -- Size the Bet. Use a staking method (Kelly Criterion or a fraction thereof, covered in later chapters) to determine the appropriate wager size relative to your bankroll and edge.
Step 6 -- Record Everything. Log the bet, your estimated probability, the odds, and the result. Over time, review your calibration: are your 60% estimates winning about 60% of the time?
Common Pitfalls
- Confusing odds formats. American odds of -110 and decimal odds of 1.10 are very different. Always confirm the format before converting.
- Ignoring the vig. Raw implied probabilities are not true probabilities. Always remove the vig before comparing to your estimates.
- Gambler's Fallacy. Past independent outcomes do not change future probabilities. A team "due for a win" has no mathematical basis.
- Treating parlays as value plays. Parlays compound the vig. Unless each individual leg has positive EV, the parlay has negative EV.
- Small sample conclusions. Fifty bets is not enough to draw meaningful conclusions about edge. Hundreds to thousands are needed.
One-Sentence Summary
Profitable sports betting reduces to one discipline: consistently identifying situations where your estimated probability of an outcome exceeds the implied probability in the odds, then wagering accordingly with proper bankroll management.