Chapter 11 Quiz: Understanding Betting Markets

Instructions: Answer all 25 questions. This quiz is worth 100 points. You have 75 minutes. A calculator is permitted; no notes or internet access. For multiple choice, select the single best answer. For code analysis, assume Python 3.10+.

Section 1: Multiple Choice (10 questions, 3 points each = 30 points)

Question 1. Which of the following best describes "Closing Line Value" (CLV)?

(A) The difference between the opening line and the closing line

(B) The profit or loss on a bet after the game concludes

(D) The vigorish embedded in the closing line

Answer

**(C) The difference between the odds you obtained at bet placement and the final closing odds.** CLV measures whether you got a better or worse price than the market's final, most-informed estimate. If you bet Team A -3 and the line closes at Team A -4.5, you achieved positive CLV of 1.5 points --- you got a better number than the market's final price. Research consistently shows that CLV is the strongest predictor of long-term betting profitability.

Question 2. A sportsbook reports that 75% of bets are on Team A, but the line moves from Team A -6 to Team A -5. This is an example of:

(A) Steam movement

(B) Reverse line movement

(D) Key number adjustment

Answer

**(B) Reverse line movement.** Reverse line movement occurs when the line moves in the opposite direction of the side receiving the majority of bets. Despite 75% of bets being on Team A at -6, the line moved toward Team B (from -6 to -5), suggesting that the money on Team B (the 25% of bets) represents larger, sharper wagers that the sportsbook respects enough to move its line against the public.

Question 3. Under the semi-strong form of the Efficient Market Hypothesis, which of the following could still provide a profitable betting edge?

(A) Historical line movement patterns

(B) Past game results and statistics

(D) Publicly available weather forecasts

Answer

**(C) Non-public information about a player's undisclosed injury.** Semi-strong form efficiency states that all publicly available information is already reflected in the odds. Historical data (A), past results (B), and public weather forecasts (D) are all publicly available and would be priced into an efficient market. Only private or non-public information --- such as knowledge of an undisclosed injury --- could provide an edge under semi-strong efficiency. Under strong-form efficiency, even this would be priced in.

Question 4. Which of the following is the primary reason NFL betting lines are "sticky" at the numbers 3 and 7?

(A) Sportsbooks prefer round numbers for display purposes

(B) These numbers correspond to the most common final margins due to the scoring structure of football

(D) Sharp bettors refuse to bet on non-key numbers

Answer

**(B) These numbers correspond to the most common final margins due to the scoring structure of football.** In the NFL, a field goal is worth 3 points and a touchdown plus extra point is worth 7 points. Historically, approximately 15% of NFL games are decided by exactly 3 points and about 9% by exactly 7 points. Because these margins are disproportionately common, the difference in cover probability between -2.5 and -3.5 (or -6.5 and -7.5) is larger than for non-key numbers, making lines resistant to moving through these numbers.

Question 5. A bettor has a CLV of +2.1% in implied probability over 800 bets on standard -110 lines. Approximately what long-term ROI should this bettor expect?

(A) +0.5%

(B) +2.1%

(D) +10.0%

Answer

**(C) +4.0%.** CLV in implied probability translates roughly to ROI through the relationship: ROI is approximately equal to CLV divided by the implied probability of the bet, multiplied by the ratio of implied probability to the vig-adjusted break-even rate. For standard -110 lines, the break-even implied probability is 52.38%. A +2.1% CLV means the bettor is effectively getting fair odds of about 54.5% on bets where the true probability is 54.5%, yielding an edge of roughly 2.1/52.38 = 4.0% ROI. The exact figure depends on the distribution of odds, but approximately 2x the CLV percentage is a reasonable estimate for standard lines.

Question 6. A "steam move" in sports betting is best described as:

(A) A single large bet that moves one sportsbook's line significantly

(B) A gradual line movement over several hours driven by public action

(D) A line movement caused by a weather-related event

Answer

**(C) Coordinated, rapid line movements across multiple sportsbooks in the same direction.** A steam move occurs when sharp action at one or more leading books triggers a cascade of line adjustments across the market. Other books move their lines even without receiving significant action themselves, either because they subscribe to automated line feeds, because they monitor sharp book movements, or because they receive copycat sharp action from bettors chasing the steam. Steam moves typically propagate within minutes.

Question 7. The favorite-longshot bias predicts that:

(A) Favorites cover the spread more often than the market implies

(B) Longshots are underpriced relative to their true probability

(D) There is no systematic relationship between odds and value

Answer

**(C) Longshots are overpriced relative to their true probability.** The favorite-longshot bias is the well-documented finding that longshot bets (those with high positive American odds or high decimal odds) tend to offer worse value than their odds suggest. The implied probabilities for longshots tend to overstate their true probability of winning, while favorites tend to be slightly underpriced. This bias has been documented across horse racing, football, basketball, and other sports, though its magnitude varies.

Question 8. Which sportsbook type typically incorporates new information into its lines the fastest?

(A) Retail-focused mobile sportsbook

(B) Sharp-focused offshore book with low limits

(D) State-licensed casino sportsbook

Answer

**(C) Market-making book that accepts large wagers from professionals.** Market-making books (such as Pinnacle in the international market) incorporate information fastest because they accept sharp action, which serves as a direct signal of new information. These books have sophisticated risk management and automated line-adjustment systems. Sharp-focused offshore books (B) also adjust quickly but often have lower limits. Retail books (A, D) typically lag because they receive less sharp action and may rely on copying market-maker lines with a delay.

Question 9. If a bettor places a bet at -110 and the closing line moves to -120 on the same side, the bettor has:

(A) Negative CLV because the line moved against them

(B) Positive CLV because they got a better price than the closing line

(D) Cannot determine CLV without knowing the game outcome

Answer

**(B) Positive CLV because they got a better price than the closing line.** The bettor bet at -110 (implied probability 52.38%) and the closing line on the same side is -120 (implied probability 54.55%). Since the bettor got a better price (lower implied probability = better odds) than the market's final assessment, they have positive CLV of approximately 2.17% in implied probability terms. CLV is independent of the game outcome --- it measures the quality of the price obtained.

Question 10. In the context of sports betting market efficiency, the "Brier score" is used to:

(A) Measure the profit generated by a betting system

(B) Evaluate the calibration and accuracy of probabilistic predictions

(D) Determine the optimal bet size using the Kelly Criterion

Answer

**(B) Evaluate the calibration and accuracy of probabilistic predictions.** The Brier score measures the accuracy of probabilistic predictions by computing the mean squared difference between predicted probabilities and actual binary outcomes. It ranges from 0 (perfect prediction) to 1 (worst possible prediction), with 0.25 representing the score of a naive 50/50 prediction for balanced events. In Chapter 11, it is used to assess whether the closing lines of a betting market are well-calibrated --- that is, whether events priced at 60% actually occur approximately 60% of the time.

Section 2: True/False (5 questions, 3 points each = 15 points)

Write "True" or "False." Full credit requires correct identification only.

Question 11. True or False: A bettor who consistently achieves positive CLV but has a losing record over 200 bets should abandon their betting strategy.

Answer

**False.** Positive CLV indicates that the bettor is getting better prices than the market's final, most-informed estimate. Over a sample of only 200 bets, variance can easily produce a losing record even for a bettor with genuine edge. CLV is a process-oriented metric, while win/loss records over small samples are heavily influenced by variance. A bettor with consistent positive CLV should maintain their strategy, as long-term results are expected to converge to positive profitability. A sample of 200 bets at standard -110 lines with a true 53% win rate has roughly a 10-15% probability of showing a negative ROI.

Question 12. True or False: If a sportsbook reports that 80% of the money is on one side of a game, the line will always move toward that side.

Answer

**False.** Sportsbooks do not mechanically move lines in response to the percentage of money alone. Other factors include: the identity of the bettors (sharp vs. recreational), the book's own liability and risk tolerance, information from other sportsbooks' line movements, and the book's assessment of the "correct" line. In cases of reverse line movement, the line moves against the side with the majority of money, typically because the book respects the quality of the money on the minority side or because they believe the public is wrong.

Question 13. True or False: In a perfectly efficient betting market, no bettor could achieve positive CLV on average over a large sample.

Answer

**True.** In a perfectly efficient market, the closing line represents the true probability of the outcome. Any bet placed before the close would have an expected CLV of zero on average, because the closing line would already incorporate all information, including the information revealed by the bettor's own wager. In practice, markets are approximately but not perfectly efficient, which is why some bettors do achieve consistent positive CLV.

Question 14. True or False: The speed at which a sportsbook adjusts its lines is independent of the size of the wagers it receives.

Answer

**False.** Bet size is a significant factor in line adjustment. A single $50,000 wager from a known sharp account will move a line much faster than 500 individual $100 bets from recreational accounts, even though the total money is the same. Sportsbooks weight the information content of bets by the bettor's track record and account classification. Large bets from sharps are treated as informative signals about the true line.

Question 15. True or False: The Brier score of a well-calibrated betting market's closing lines should be close to 0.

Answer

**False.** A Brier score of 0 would mean perfect prediction --- knowing the outcome of every game with certainty. Sports outcomes are inherently uncertain, so even perfectly calibrated probabilities will produce a Brier score well above 0. For NFL games, a well-calibrated market typically produces Brier scores around 0.20-0.24. A Brier score of 0.25 corresponds to the naive baseline of predicting 50% for every game. "Close to 0" would imply near-certain prediction, which is impossible in sports.

Section 3: Fill in the Blank (3 questions, 4 points each = 12 points)

Question 16. A bettor who is consistently limited or banned by sportsbooks is most likely classified as a __________ bettor, because sportsbooks view their action as __________ rather than as a source of __________.

Answer

A **sharp** bettor, because sportsbooks view their action as **informed / a liability** rather than as a source of **revenue / profit**. Sharp bettors are limited or banned because their wagers consistently provide the sportsbook with negative expected value. Unlike recreational bettors, whose action generates vig-based profit, sharp bettors extract money from the sportsbook over time. Books tolerate sharp action to a point (it provides price-discovery information), but eventually limit accounts that consistently beat the closing line.

Question 17. The __________ __________ bias is the empirical finding that bets on outcomes with low probability tend to be __________ relative to their true likelihood, while bets on likely outcomes tend to be __________.

Answer

The **favorite-longshot** bias is the empirical finding that bets on outcomes with low probability tend to be **overpriced** relative to their true likelihood, while bets on likely outcomes tend to be **underpriced / fairly priced**. The favorite-longshot bias means that the market systematically overstates the chances of longshots and slightly understates the chances of favorites. This pattern has been documented across many sports and is attributed to a combination of risk-seeking behavior by recreational bettors and systematic probability misperception.

Question 18. When multiple sportsbooks simultaneously move their lines on the same game in the same direction within a short time window, this coordinated movement is called a __________ __________, and it is typically initiated by __________ action at __________ books.

Answer

A **steam move**, and it is typically initiated by **sharp** action at **market-making / leading** books. Steam moves propagate through the market as other books observe the movement at sharp/market-making books and adjust their own lines accordingly, either through automated line feeds or manual adjustment. The speed of propagation has increased dramatically with technology, reducing the window in which bettors can exploit the lagging books.

Section 4: Short Answer (3 questions, 5 points each = 15 points)

Answer each question in 3-5 sentences.

Question 19. Explain why the closing line is generally considered a more accurate predictor of game outcomes than the opening line. What information is incorporated between open and close?

Answer

The closing line reflects the cumulative information revealed through the entire life of the market, making it significantly more informed than the opening line. Between open and close, multiple types of information are incorporated: sharp bettor opinions (expressed through their wagers), injury reports and lineup confirmations, weather updates, public betting patterns, and cross-market signals from other sportsbooks and betting exchanges. The process of price discovery operates as sharp bettors identify mispricings and bet into them, moving the line toward its "true" value. Research by Pinnacle and academic studies have shown that closing lines are virtually unbiased estimators of game outcomes. Teams priced at 60% implied probability at the close win approximately 60% of the time over large samples. This calibration is the result of an efficient market aggregating diverse information sources through the wagering process.

Question 20. A recreational bettor argues: "CLV doesn't matter because you can't deposit CLV into your bank account --- only winning bets pay." Refute this argument with a clear explanation of why CLV is a superior long-term metric.

Answer

While it is true that only winning bets generate direct profit, CLV is the best available predictor of whether a bettor will be profitable over the long run. In the short term, any bettor can get lucky or unlucky regardless of the quality of their bets. Win rate over small samples is dominated by variance and is an unreliable indicator of skill. CLV, by contrast, measures the quality of each bet at the time it is placed, comparing it against the market's final, most-informed assessment. Consider an analogy: a poker player who consistently gets money in with the best hand (positive expected value) but loses several consecutive hands is still playing well. The short-term results are noise; the quality of the decisions is the signal. Similarly, a bettor who consistently beats the closing line is making bets that the market's final wisdom deems to have been underpriced. Over thousands of bets, this edge compounds into profit. Research consistently shows that bettors with positive CLV are profitable over sufficient sample sizes, while those with negative CLV are not, regardless of their short-term win rates.

Question 21. Describe two specific market inefficiencies that have been documented in sports betting research. For each, explain why the inefficiency exists and whether it has persisted or been corrected over time.

Answer

**Inefficiency 1: The Favorite-Longshot Bias.** Longshot bets (high-odds outcomes) consistently offer negative expected value relative to their true probability, while favorites are closer to fairly priced. This bias exists because recreational bettors overweight the excitement of large payoffs and systematically overestimate the probability of unlikely outcomes, while sportsbooks exploit this demand by inflating longshot prices. The bias has persisted for decades across multiple sports and markets, though its magnitude has decreased in liquid markets where sharp action competes against recreational pricing. **Inefficiency 2: Home Underdog Bias in NFL.** Multiple studies have documented that NFL home underdogs have historically covered the spread at a rate above 50%, particularly when receiving large spreads. This inefficiency may exist because public bettors are attracted to road favorites (brand-name teams in primetime), creating systematic mispricing of home underdogs. However, this bias has weakened significantly in recent years as the information has become widely publicized and sportsbooks have adjusted their models accordingly, illustrating how documented inefficiencies tend to diminish once they are publicly known.

Section 5: Code Analysis (2 questions, 6 points each = 12 points)

Question 22. Examine the following Python function:

def calculate_clv(bet_odds: int, closing_odds: int, bet_side: str) -> dict:
    """Calculate CLV for a spread/total bet given American odds."""
    def implied_prob(odds: int) -> float:
        if odds > 0:
            return 100 / (odds + 100)
        else:
            return abs(odds) / (abs(odds) + 100)

    bet_prob = implied_prob(bet_odds)
    close_prob = implied_prob(closing_odds)

    if bet_side == "favorite":
        clv = close_prob - bet_prob
    else:
        clv = bet_prob - close_prob

    return {
        "bet_implied": round(bet_prob, 4),
        "close_implied": round(close_prob, 4),
        "clv": round(clv, 4),
        "positive_clv": clv > 0
    }

(a) Trace through calculate_clv(-110, -130, "favorite") and write the exact return value.

(b) There is a conceptual flaw in how CLV is computed for the two sides. Explain the flaw and propose a fix.

(c) Explain why using implied probability (with vig included) to compute CLV can be misleading and how you would modify the function to use no-vig probabilities.

Answer

**(a)** Tracing `calculate_clv(-110, -130, "favorite")`: - `bet_prob = implied_prob(-110) = 110 / (110 + 100) = 110 / 210 = 0.52381` - `close_prob = implied_prob(-130) = 130 / (130 + 100) = 130 / 230 = 0.56522` - `bet_side == "favorite"`, so `clv = 0.56522 - 0.52381 = 0.04141` Return value:

{"bet_implied": 0.5238, "close_implied": 0.5652, "clv": 0.0414, "positive_clv": True}

**(b)** The conceptual flaw is that CLV should always be computed the same way regardless of which side the bettor is on: **CLV = closing_implied_on_your_side - your_implied_probability**. The current logic reverses the calculation for the underdog side. A simpler and correct approach is: if the closing line on your side has moved to a higher implied probability than what you bet at, you have positive CLV (the market now says your bet was underpriced). The function should just compute `clv = close_prob - bet_prob` for all cases, assuming both `bet_odds` and `closing_odds` are the odds on the side the bettor bet on. **(c)** Using vig-inclusive implied probabilities inflates both the bet and closing probabilities. To use no-vig probabilities, you would need the odds on both sides of the market at both the time of bet and at close. Remove the vig using the multiplicative method: `no_vig_prob = implied_prob / overround`, where `overround = implied_prob_side_a + implied_prob_side_b`. This gives a fairer measure of the true edge.

Question 23. Examine the following Python code that simulates a market efficiency test:

import numpy as np

def market_efficiency_test(n_games: int = 1000, seed: int = 42) -> dict:
    rng = np.random.default_rng(seed)
    true_probs = rng.uniform(0.35, 0.75, n_games)
    noise = rng.normal(0, 0.02, n_games)
    market_probs = np.clip(true_probs + noise, 0.01, 0.99)
    outcomes = rng.binomial(1, true_probs)

    bins = np.arange(0.35, 0.80, 0.05)
    bin_indices = np.digitize(market_probs, bins) - 1

    results = []
    for i in range(len(bins) - 1):
        mask = bin_indices == i
        if mask.sum() > 0:
            bin_mid = (bins[i] + bins[i + 1]) / 2
            actual_rate = outcomes[mask].mean()
            results.append({
                "bin": f"{bins[i]:.2f}-{bins[i+1]:.2f}",
                "midpoint": round(bin_mid, 3),
                "n_games": int(mask.sum()),
                "actual_rate": round(actual_rate, 3)
            })

    brier = np.mean((market_probs - outcomes) ** 2)
    return {"bins": results, "brier_score": round(brier, 4)}

(a) Explain what this simulation is testing and how it generates synthetic data.

(b) The noise parameter rng.normal(0, 0.02, n_games) represents market imperfection. If you increased the standard deviation from 0.02 to 0.10, how would the Brier score change and why?

(c) What does a Brier score represent, and what range of values would indicate a well-calibrated market in this simulation?

Answer

**(a)** The simulation tests whether a synthetic betting market's closing probabilities are well-calibrated. It generates true probabilities uniformly between 35% and 75%, then adds Gaussian noise (mean 0, standard deviation 2%) to create "market probabilities" that approximate but do not perfectly match the true probabilities. Outcomes are simulated as Bernoulli trials using the true probabilities. The market probabilities are then binned in 5% increments, and the actual win rate within each bin is compared to the bin midpoint. If the market is well-calibrated, games priced around 60% should win approximately 60% of the time. **(b)** Increasing the noise standard deviation from 0.02 to 0.10 would worsen (increase) the Brier score. The Brier score measures the mean squared error between predictions and outcomes. With larger noise, the market probabilities deviate more from the true probabilities, leading to larger prediction errors on average. The Brier score would increase from approximately 0.22-0.23 to a higher value reflecting the additional noise-induced miscalibration. **(c)** The Brier score is the mean squared difference between predicted probabilities and binary outcomes, ranging from 0 (perfect) to 1 (worst). For this simulation with true probabilities between 0.35 and 0.75, a well-calibrated market (low noise) should produce a Brier score in the range of approximately 0.20-0.24. This is close to the inherent uncertainty of the events and cannot reach 0 because the outcomes are genuinely probabilistic. A Brier score significantly above 0.25 would indicate poor calibration.

Section 6: Applied Problems (2 questions, 8 points each = 16 points)

Question 24. You are analyzing line movement data for an NFL Sunday with 14 games. You have the following summary:

Metric	Value
Games with > 1 point of line movement	6
Games with reverse line movement	3
Average opening spread	-4.2
Average closing spread	-4.5
Games where sharp money identified on underdog	5
Games where sharp money identified on favorite	4
Games where no clear sharp signal	5

Additionally, for the 9 games with identified sharp signals:

Sharp Side	Games	ATS Record	CLV if bet at open
Sharp on underdog	5	3-2	+0.8 pts avg
Sharp on favorite	4	2-2	+0.5 pts avg

(a) (2 points) Calculate the overall ATS record for games with a sharp signal and compare it to the standard break-even rate. Is the difference statistically significant for this sample?

(b) (2 points) Calculate the average CLV for following the sharp side at the opening line across all 9 games. What does this suggest about the predictive value of sharp money signals?

(c) (2 points) Three of the 14 games showed reverse line movement. If you had blindly bet the RLM side in all three games, and the ATS record was 2-1, calculate the ROI assuming standard -110 odds and $100 bets.

(d) (2 points) Critically evaluate using a single NFL Sunday (14 games) to draw conclusions about the profitability of sharp money signals or RLM. What minimum sample size would you recommend and why?

Answer

**(a)** Overall sharp-side ATS record: 3-2 (underdogs) + 2-2 (favorites) = **5-4 (55.6%)**. Break-even rate at -110: 110/210 = 52.38%. The difference: 55.6% - 52.38% = 3.2 percentage points above break-even. For a sample of 9 games, the standard error of a proportion at 50% is sqrt(0.5 x 0.5 / 9) = 0.167 = 16.7%. A z-test yields z = (0.556 - 0.524) / 0.167 = 0.19. The p-value is approximately 0.42, which is **not statistically significant**. The sample is far too small to draw any conclusions. **(b)** Average CLV: (5 games x +0.8 pts + 4 games x +0.5 pts) / 9 = (4.0 + 2.0) / 9 = **+0.67 points per game**. Positive average CLV suggests that sharp money signals do contain predictive information --- betting the sharp side at the open consistently yields prices better than the close. However, the sample of 9 games is insufficient for statistical significance. Over larger samples, even small positive CLV compounds into meaningful profit. **(c)** RLM side: 2-1 at -110, $100 bets. - 2 wins: 2 x $90.91 = $181.82 profit - 1 loss: 1 x $100 = $100 loss - Net profit: $181.82 - $100 = $81.82 - Total wagered: 3 x $100 = $300 - ROI: $81.82 / $300 = **+27.3%** **(d)** A single 14-game NFL Sunday is entirely inadequate for evaluating any betting strategy. With only 9 sharp-signal games and 3 RLM games, the confidence intervals are enormous. To detect a 3% edge above the 52.38% break-even rate with 80% power at the 5% significance level, the required sample size is approximately n = (z_alpha + z_beta)^2 x p(1-p) / delta^2 = (1.96 + 0.84)^2 x 0.524 x 0.476 / 0.03^2 = approximately **1,740 bets**. At 14 games per week and 18 weeks per season, this represents roughly 7 full NFL seasons. The recommendation is a minimum of **500-1,000 bets** for preliminary conclusions and **1,500+** for robust statistical inference.

Question 25. A bettor provides you with their complete betting record for the past NFL season. The summary statistics are:

Metric	Value
Total bets	320
Win rate	53.4%
Average odds	-108
ROI	+3.2%
Average CLV (implied probability)	+1.6%
CLV hit rate (% of bets with positive CLV)	57%
Bets on spreads	180
Bets on totals	90
Bets on moneylines	50

(a) (2 points) Calculate the break-even win rate at average odds of -108 and determine whether the bettor's 53.4% win rate is significantly above break-even at the 95% confidence level.

(b) (2 points) Analyze the bettor's CLV profile. With an average CLV of +1.6% and a CLV hit rate of 57%, is this consistent with a skilled bettor or could it be explained by luck? Calculate the expected CLV hit rate for a random bettor and compare.

(c) (2 points) The bettor claims their edge is entirely in totals, where they have a 58% win rate on 90 bets. Calculate the 95% confidence interval for the true win rate on totals and assess whether this claim is statistically supported.

(d) (2 points) Based on all available evidence, write a 3-4 sentence evaluation of this bettor's skill level and the sustainability of their results.

Answer

**(a)** Break-even win rate at -108: 108 / (108 + 100) = 108 / 208 = **51.92%**. Observed win rate: 53.4%. Difference: 1.48 percentage points. Standard error: sqrt(0.534 x 0.466 / 320) = sqrt(0.000778) = 0.0279 = 2.79%. z-score: (0.534 - 0.5192) / 0.0279 = 0.53. The p-value is approximately 0.30, which is **not statistically significant at the 95% level**. The 95% confidence interval for the true win rate is 53.4% +/- 5.5%, or approximately 47.9% to 58.9%. This interval includes the break-even rate, so we cannot conclude the bettor is skilled based on win rate alone. **(b)** A random bettor should have a CLV hit rate of approximately 50% (equally likely to be above or below the closing line by chance). The observed 57% CLV hit rate over 320 bets has a standard error of sqrt(0.5 x 0.5 / 320) = 2.80%. The z-score is (0.57 - 0.50) / 0.028 = 2.50, yielding a p-value of approximately 0.006. This is **statistically significant** at the 1% level. Combined with the +1.6% average CLV, this provides strong evidence that the bettor is obtaining systematically better prices than the closing line, which is a hallmark of skilled betting. The CLV evidence is more convincing than the win rate evidence. **(c)** Win rate on totals: 58% on 90 bets. Standard error: sqrt(0.58 x 0.42 / 90) = sqrt(0.00271) = 0.0521 = 5.21%. 95% confidence interval: 58% +/- 1.96 x 5.21% = 58% +/- 10.2% = **47.8% to 68.2%**. This interval includes the break-even rate of 51.92%, so while the point estimate looks impressive, the sample of 90 bets is too small to conclusively support the claim that the bettor has a genuine edge specifically in totals. Approximately 200+ total bets would be needed to narrow the confidence interval sufficiently. **(d)** This bettor shows promising signs of genuine skill. The strongest evidence is the CLV profile: a 57% CLV hit rate over 320 bets is statistically significant and indicates the bettor is systematically identifying value before the market fully prices it in. The +3.2% ROI is healthy but not yet statistically distinguishable from zero given the sample size. The claim of a specific totals edge cannot be confirmed with only 90 bets. Overall, the bettor should continue tracking CLV as the primary diagnostic and aim for 500+ bets before drawing firm conclusions about their long-term profitability.

Scoring Summary

Section	Questions	Points Each	Total
1. Multiple Choice	10	3	30
2. True/False	5	3	15
3. Fill in the Blank	3	4	12
4. Short Answer	3	5	15
5. Code Analysis	2	6	12
6. Applied Problems	2	8	16
Total	25	---	100

Grade Thresholds

Grade	Score Range	Percentage
A	90-100	90-100%
B	80-89	80-89%
C	70-79	70-79%
D	60-69	60-69%
F	0-59	0-59%