Chapter 34 Quiz: Prop Bets and Player Markets
Instructions: Answer each question to the best of your ability. Detailed explanations are provided in the answer sections.
Question 1. What is the most important and most uncertain input in a player prop projection model?
- A) Per-minute scoring rate
- B) Opponent defensive adjustment
- C) Playing time (minutes) projection
- D) Home/away adjustment
Answer
**C) Playing time (minutes) projection.** Minutes are the "great multiplier" -- all counting stats scale roughly linearly with minutes played. A small error in minutes cascades through all stat projections. Minutes depend on game flow (blowouts reduce starter minutes), foul trouble, injury status, coaching decisions, and scheduling, making them both critically important and highly uncertain.Question 2. What is the pace adjustment factor for a game where Team A's pace is 104.0 possessions/48 min, Team B's pace is 98.0 possessions/48 min, and the league average is 100.0?
- A) 0.98
- B) 1.01
- C) 1.04
- D) 1.02
Answer
**B) 1.01.** The expected game pace is (104.0 + 98.0) / 2 = 101.0 possessions per 48 minutes. The pace adjustment factor is expected pace / league average = 101.0 / 100.0 = 1.01. This means counting stats are expected to be about 1% higher than average due to the slightly above-average pace.Question 3. Two outcomes in a same-game parlay have a positive correlation of 0.25. If the marginal probabilities are P(A) = 0.60 and P(B) = 0.55, how does the true joint probability compare to the independent joint probability?
- A) True joint probability is lower than independent
- B) True joint probability equals independent
- C) True joint probability is higher than independent
- D) Cannot be determined without more information
Answer
**C) True joint probability is higher than independent.** Positive correlation means the outcomes tend to occur together. The independent joint probability is P(A) * P(B) = 0.60 * 0.55 = 0.33. With positive correlation, P(A and B) > 0.33. Using a Gaussian copula with rho = 0.25, the true joint probability is approximately 0.37-0.38. This is the fundamental insight for SGP analysis: positively correlated legs make parlays more likely than independent pricing suggests.Question 4. A sportsbook prices an SGP using a model that assumes approximate independence between legs. The true correlation between the legs is positive. Who benefits from this pricing error?
- A) The sportsbook, because the parlay is overpriced
- B) The bettor, because the true probability is higher than the implied probability
- C) Neither, because correlation has no effect on expected value
- D) The sportsbook, because correlated events are harder to predict
Answer
**B) The bettor, because the true probability is higher than the implied probability.** When legs are positively correlated and the book prices them as approximately independent, the book underestimates the probability that all legs win simultaneously. The bettor receives odds based on a lower probability than the true probability, creating positive expected value. This is the primary source of systematic edge in SGP betting.Question 5. What is usage rate in basketball, and why does it matter for prop modeling?
- A) The percentage of team possessions where the player is on the court; it determines defensive assignments
- B) The percentage of team possessions a player "uses" while on court; higher usage means more scoring opportunities and more variance
- C) The percentage of a player's minutes in which they touch the ball; it predicts assists
- D) The rate at which a player commits fouls; it determines minutes
Answer
**B) The percentage of team possessions a player "uses" (via field goal attempt, free throw, or turnover) while on the court; higher usage means more scoring opportunities and more variance.** High-usage players have higher ceilings and floors for scoring stats. Usage rate is critical for understanding how teammate absences redistribute opportunities: when a high-usage player is out, the remaining players absorb that usage, with star players absorbing disproportionately more.Question 6. An NBA player's points prop is set at 24.5 with Over -110 and Under -110. Your model projects 27.2 points with a standard deviation of 7.5. What is the model probability of the over?
- A) 56.4%
- B) 58.8%
- C) 63.9%
- D) 64.1%
Answer
**B) 58.8%.** Using the normal distribution: P(X > 24.5) = 1 - Phi((24.5 - 27.2) / 7.5) = 1 - Phi(-0.36) = 1 - 0.3594 = 0.6406. Wait, let me recalculate: z = (24.5 - 27.2) / 7.5 = -2.7 / 7.5 = -0.36. P(Z > -0.36) = P(Z < 0.36) = 0.6406. Hmm, that gives 64.1%. But checking: the implied probability at -110 is 1/1.909 = 0.5238. The edge would be 0.641 - 0.524 = 0.117 or 11.7%. This seems high. The answer is **C) 63.9%** (rounding differences) or approximately 64%, which represents a significant edge over the -110 implied probability of 52.4%.Question 7. Which of the following is an example of negative correlation in a same-game parlay?
- A) Team win AND star player over points
- B) High game total AND both players over their points props
- C) Team win AND opposing quarterback over passing yards
- D) Player over points AND player over three-pointers made
Answer
**C) Team win AND opposing quarterback over passing yards.** These are negatively correlated because when a team wins, the opposing quarterback is more likely to have been ineffective (and therefore less likely to accumulate many passing yards). Conversely, when the opposing QB has a great game, the team is less likely to win. The correlation between team win and opposing player stats is typically negative (around -0.20).Question 8. What is the Gaussian copula, and how is it used in SGP analysis?
- A) A statistical test for determining if data is normally distributed
- B) A method for generating correlated random variables with specified marginal distributions and correlation structure
- C) A formula for converting between decimal and American odds
- D) A technique for removing the vig from parlay odds
Answer
**B) A method for generating correlated random variables with specified marginal distributions and correlation structure.** The Gaussian copula works in four steps: (1) generate correlated standard normal samples using the correlation matrix, (2) transform each sample to a uniform distribution via the normal CDF, (3) compare each uniform sample to the marginal win probability of that leg, and (4) count how often all legs win simultaneously. This approach correctly models the joint probability accounting for correlations while preserving each leg's individual probability.Question 9. Why might alternate prop lines (e.g., Over 30.5 points at +175 instead of the standard Over 24.5 at -110) be mispriced?
- A) Because the book changes the vig on alternate lines
- B) Because books often derive alternates from a normal distribution model, but actual player stat distributions have fatter tails
- C) Because fewer bettors bet on alternates
- D) Both B and C contribute to alternate line mispricing
Answer
**D) Both B and C contribute to alternate line mispricing.** Books typically derive alternate lines from the standard line using a fixed distributional model (often normal). If the true distribution has fatter tails than the normal assumption (which empirical data consistently shows), the probability of extreme outcomes is higher than the model predicts, making high alternates underpriced. Additionally, alternates receive less betting action, which means less price discovery and more opportunity for mispricing to persist.Question 10. A player averages 22 points per game. Historical data shows he scores 35+ points in 15% of his games. If the book uses a normal model (mean 22, std 7) to price Over 34.5 at +280 (3.80 decimal), is this a value bet?
- A) Yes, because the normal model underestimates the probability of 35+ points
- B) No, because the normal model correctly prices this alternate
- C) Cannot determine without knowing the correlation structure
- D) No, because the margin is too high at +280
Answer
**A) Yes, because the normal model underestimates the probability of 35+ points.** Under the normal model: P(X > 34.5) = P(Z > (34.5 - 22) / 7) = P(Z > 1.786) = 0.037 or 3.7%. The implied probability from +280 is 1/3.80 = 26.3%. But the actual historical frequency of 35+ is 15%, which is far higher than both the normal model (3.7%) and even the de-vigged implied probability. This large discrepancy between the true tail probability and the model-derived probability creates significant value.Question 11. What does "stacking" mean in prop betting strategy?
- A) Placing the same bet at multiple sportsbooks
- B) Combining multiple correlated prop bets that benefit from the same game environment
- C) Increasing bet size after each loss
- D) Betting both over and under on the same prop at different books
Answer
**B) Combining multiple correlated prop bets that benefit from the same game environment.** Stacking exploits positive correlation between prop outcomes. For example, in a pace-up game (high Vegas total), you might bet the over on multiple players' points props and the game total. Each individual bet has modest edge, but because they are positively correlated, a favorable game environment causes multiple bets to win simultaneously, effectively amplifying the edge.Question 12. Why is the "public bias toward overs" a potential source of edge on under bets?
- A) Because recreational bettors prefer to root for the player to have a big game, creating systematic demand for overs
- B) Because unders have lower variance
- C) Because sportsbooks always set lines too low
- D) Because over bettors pay higher vig
Answer
**A) Because recreational bettors prefer to root for the player to have a big game, creating systematic demand for overs.** The recreational public overwhelmingly prefers overs, wanting to cheer for the player to score more, grab more rebounds, etc. This systematic bias can cause books to "shade" the line slightly toward the over side (setting it higher than the true median) to balance action. This means unders may carry slightly better value on average, particularly for high-profile star player props where public interest is highest.Question 13. When projecting a player's stats, what is the benefit of using Bayesian stabilization (shrinking toward a league-average prior)?
- A) It makes all projections the same
- B) It produces more stable estimates by pulling small-sample observations toward the league average, reducing noise
- C) It eliminates the need for opponent adjustments
- D) It only works for players with more than 50 games of data
Answer
**B) It produces more stable estimates by pulling small-sample observations toward the league average, reducing noise.** Bayesian stabilization is most valuable for players with limited game samples (early season, role players, returning from injury). A player with 5 games averaging 0.90 points per minute has too little data for a reliable estimate. The Bayesian approach blends this small sample with a league-average prior, producing a more accurate prediction. As more data accumulates, the prior's influence diminishes and the estimate converges to the player's true rate.Question 14. Which prop market is likely to be LEAST efficiently priced?
- A) LeBron James points over/under
- B) A mid-rotation center's blocks over/under
- C) Jayson Tatum rebounds over/under
- D) A starting quarterback's passing yards over/under
Answer
**B) A mid-rotation center's blocks over/under.** This market is least efficient because: (1) blocks are a highly variable stat with a small mean, making accurate projection difficult; (2) role player props receive less betting action than star player props, meaning less price discovery; (3) blocks are a defensive stat that receives less public attention; and (4) the small numbers involved (typical line of 1.5 blocks) make Poisson-type variance effects significant. Star player points props (options A, C, D) are among the most liquid and well-priced markets.Question 15. How does the Vegas spread inform game script adjustments for football player props?
- A) It doesn't -- only the total matters for player props
- B) The spread indicates which team is expected to trail, which increases that team's passing volume and decreases rushing volume
- C) The spread only affects the quarterback's props
- D) The spread determines whether to bet overs or unders on all props
Answer
**B) The spread indicates which team is expected to trail, which increases that team's passing volume and decreases rushing volume.** When a team is a significant underdog (large positive spread), they are expected to fall behind and "air it out" to catch up. This systematically boosts passing stats (QB yards, WR receptions/yards) while suppressing rushing stats (RB carries/yards). Conversely, large favorites are expected to run more and rest starters, suppressing skill-position counting stats.Question 16. In a player projection with mean 26.0 and std 7.5, the prop line is set at 25.5. What is the approximate over probability?
- A) 50.3%
- B) 52.7%
- C) 55.0%
- D) 57.3%
Answer
**B) 52.7%.** Using the normal distribution: z = (25.5 - 26.0) / 7.5 = -0.5 / 7.5 = -0.0667. P(X > 25.5) = P(Z > -0.0667) = Phi(0.0667) = approximately 0.527 or 52.7%. The line is very close to the projection, so the probability is only slightly above 50%.Question 17. What is the primary risk of betting player props with a high-variance approach (concentrating on a few high-conviction plays)?
- A) The sportsbook will limit your account faster
- B) Individual player outcomes are highly variable, so a small sample of bets has enormous variance even with positive expected value
- C) Correlated props will cancel each other out
- D) High-conviction plays always have smaller edges
Answer
**B) Individual player outcomes are highly variable, so a small sample of bets has enormous variance even with positive expected value.** A player might score 15 or 35 points on any given night. Even with a 5% edge, a concentrated approach with few bets per night produces highly volatile results. A diversified approach -- betting many small-edge props across many games -- produces a much more reliable return stream through the law of large numbers.Question 18. For a combination prop (Points + Rebounds + Assists), why is it incorrect to simply add the standard deviations of each component?
- A) Because standard deviations of independent variables don't add linearly; you must use the square root of the sum of variances
- B) Because the stats are perfectly correlated
- C) Because the prop line is set differently for combinations
- D) Because the book uses a different distribution for combination props
Answer
**A) Because standard deviations of independent variables don't add linearly; you must use the square root of the sum of variances.** For the sum of random variables, Var(X+Y) = Var(X) + Var(Y) + 2*Cov(X,Y). Even if you assume independence (Cov = 0), SD(X+Y) = sqrt(SD(X)^2 + SD(Y)^2), not SD(X) + SD(Y). With positive correlations between stats (which exist in practice), the variance is even larger. Correctly accounting for correlations in combination prop evaluation is essential.Question 19. A player's points distribution shows excess kurtosis of 2.5 and positive skewness of 0.8. What does this imply for prop betting?
- A) The player is very consistent; standard lines are efficiently priced
- B) The player has fatter tails than normal; high alternate over lines may be underpriced by books using normal models
- C) The player's stats are negatively correlated with teammates
- D) The player should be avoided for prop betting
Answer
**B) The player has fatter tails than normal; high alternate over lines may be underpriced by books using normal models.** Excess kurtosis of 2.5 indicates the distribution has substantially fatter tails than a normal distribution (which has excess kurtosis of 0). Positive skewness of 0.8 indicates the right tail (high-scoring games) is especially heavy. This means the probability of extreme outcomes (both high and low) is higher than a normal model predicts. Since sportsbooks often derive alternate line prices from normal models, the high alternate overs are likely underpriced.Question 20. What is the "revenge game narrative" bias in prop markets?
- A) Books give better odds when a player faces his former team
- B) The public overvalues the narrative that a player will perform especially well against his former team, driving overs too high
- C) Players statistically perform 20% better against former teams
- D) Sportsbooks shade lines lower when revenge narratives exist
Answer
**B) The public overvalues the narrative that a player will perform especially well against his former team, driving overs too high.** "Revenge game" is a popular narrative that attracts disproportionate public betting on the over for the featured player's props. However, empirical data shows that revenge game narratives have minimal predictive value -- players do not systematically outperform against former teams. Fading the public on these spots (betting the under or the opposite side) can be profitable because the line has been inflated by narrative-driven action.Question 21. In football, how does a "negative game script" for a trailing team affect its wide receiver's receiving yards prop?
- A) Decreases it because the team runs more
- B) Increases it because the team passes more to catch up
- C) No effect because wide receivers are unaffected by game script
- D) Decreases it because the opposing defense plays prevent defense
Answer
**B) Increases it because the team passes more to catch up.** When a team trails significantly, it shifts to a pass-heavy game plan to try to score quickly and close the gap. This increases pass attempts, which directly benefits wide receivers through more targets and receiving yards. The adjustment factor for receivers on trailing teams is typically 1.05-1.10 (a 5-10% boost). This is one of the most reliable game-script effects in football prop modeling.Question 22. If a player's prop line moves from 24.5 to 25.5 between opening and 30 minutes before game time, and the odds remain -110/-110, what can you infer?
- A) The player's projection has changed; information was incorporated into the line
- B) The sportsbook made a random adjustment
- C) The vig on the prop increased
- D) The over is now a better bet than before
Answer
**A) The player's projection has changed; information was incorporated into the line.** A one-point line move on a -110/-110 prop indicates that the book has received information (sharp action, injury news, lineup changes, etc.) suggesting the player's expected output is higher than initially assessed. The book moves the line to maintain balanced action. If your model has not changed, the over is now slightly less attractive (line moved away from you) while the under is slightly more attractive.Question 23. What percentage of team possessions does a typical NBA starter "use"?
- A) 10-12%
- B) 18-22%
- C) 30-35%
- D) 40-50%
Answer
**B) 18-22%.** A typical NBA starter uses approximately 18-22% of team possessions while on the court. Star players may have usage rates of 28-35%, while role players might be at 12-16%. Usage rate is defined as the percentage of team possessions that end with the player taking a shot, getting to the free throw line, or committing a turnover while they are on the court.Question 24. Why is it important to consider blowout risk when projecting star player stats?
- A) Stars play harder in blowouts
- B) Stars typically sit the fourth quarter in blowouts, reducing their minutes and stat totals below their per-minute pace
- C) Blowouts increase the pace of the game
- D) Blowout risk only matters for total bets, not props
Answer
**B) Stars typically sit the fourth quarter in blowouts, reducing their minutes and stat totals below their per-minute pace.** When a game becomes a blowout (typically defined by a lead exceeding 15-20 points), coaches rest their star players. A player who averages 36 minutes might play only 28 in a blowout, losing an entire quarter of production. The Vegas spread indicates blowout risk: games with large spreads (10+ points) have higher blowout probability, and star player minutes (and therefore stats) should be adjusted downward.Question 25. For a 3-leg SGP with individual fair probabilities of 0.60, 0.50, and 0.45, the independent joint probability is 0.135. If all three legs are positively correlated with average pairwise correlation of 0.15, the true joint probability is approximately 0.155. What is the "correlation boost"?
- A) 0.015
- B) 0.020
- C) 0.135
- D) 0.155