Chapter 12 Quiz: Line Shopping and Odds Optimization

Instructions: Answer all 25 questions. This quiz is worth 100 points. You have 75 minutes. A calculator is permitted; no notes or internet access.

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

Question 1. A bettor has accounts at three sportsbooks. For a given NFL spread bet, the odds are: Book A: -108, Book B: -110, Book C: -112. The implied probability difference between the best and worst option is approximately:

(A) 0.5% (B) 1.0% (C) 1.8% (D) 3.5%

Question 2. Line shopping is MOST valuable in which of the following situations?

(A) Betting on a 50/50 proposition at one book

(B) Betting on a heavy -300 favorite where all books agree

(C) Betting NFL player props where odds vary significantly across books

(D) Betting on the Super Bowl winner three months before the game

Question 3. The break-even win rate for a standard -110 bet is 52.38%. At reduced juice of -105, the break-even win rate is:

(A) 50.00% (B) 51.22% (C) 52.38% (D) 51.85%

Question 4. A bettor consistently shops lines and gets an average of -107 instead of the standard -110. Over 500 bets at $100 each with a 53% win rate, the additional profit from shopping is approximately:

(A) $250 (B) $750 (C) $1,500 (D) $3,000

Question 5. Which of the following is the PRIMARY reason that live betting markets show larger discrepancies between sportsbooks than pre-game markets?

(A) Sportsbooks deliberately set different live lines to attract action

(B) Live odds update at different speeds across books due to varying models and latency

(C) There is no vig in live betting

(D) Live betting is unregulated

Question 6. A bettor identifies a pure arbitrage: Book A offers Team X at +105, Book B offers Team Y at +105 (same game, opposite sides). The guaranteed profit percentage is:

(A) 2.44% (B) 5.00% (C) 10.00% (D) There is no profit

Question 7. The "middle" in sports betting refers to:

(A) The average of the opening and closing line

(B) A situation where a bettor has bets on both sides at different numbers and both can win

(C) The median odds across all sportsbooks

(D) The vig-free midpoint of a two-way market

Question 8. A bettor's "line shopping efficiency" is 65%. This means:

(A) They win 65% of their bets

(B) They capture 65% of the available improvement between worst and best available odds

(C) They use 65% of their bankroll for line shopping

(D) 65% of their profit comes from line shopping

Question 9. Which factor is LEAST likely to cause odds discrepancies between sportsbooks?

(A) Different vig structures across books

(B) Varying exposure to sharp action

(C) The color of the sportsbook's website

(D) Different models used for setting lines

Question 10. The most effective line shopping strategy for a recreational bettor who bets 5 games per week is:

(A) Opening accounts at 20+ sportsbooks

(B) Maintaining 3-5 accounts and checking all before each bet

(C) Using a single sportsbook with the lowest vig

(D) Waiting until the closing line to place all bets

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

Question 11. True or False: Line shopping provides value even for bettors who have no predictive model or betting edge.

Question 12. True or False: Arbitrage opportunities between sportsbooks are common and easy to exploit in modern regulated markets.

Question 13. True or False: A bettor who consistently gets -105 juice instead of -110 has an advantage equivalent to approximately 1.2% in implied probability per bet.

Question 14. True or False: The best time to shop lines is always as early as possible after lines are released.

Question 15. True or False: Line shopping across only two sportsbooks captures most of the theoretical benefit of shopping across ten or more books.

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

Question 16. The break-even win rate at American odds of -X is calculated as X / (X + ______), and at reduced juice of -105 this equals approximately ______%.

Question 17. When two sportsbooks offer different spreads on the same game and a bettor bets both sides to potentially win both bets if the final margin lands between the two numbers, this is called a __________.

Question 18. A bettor's __________ __________ measures what fraction of the available odds improvement (between worst and best book) they actually captured on each bet.

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

Question 19. Explain why player prop markets typically offer more line shopping value than main market spreads. Identify at least three structural factors.

Question 20. A bettor argues: "I only bet at one sportsbook because I get loyalty rewards worth $500 per year." The bettor places 400 bets per year at $100 each. Analyze whether the loyalty rewards justify foregoing line shopping. Assume line shopping saves an average of 1% in implied probability per bet.

Question 21. Describe the difference between a "middle" and an "arbitrage" in sports betting. For each, provide an example, explain the risk/reward profile, and identify when a line shopper might encounter each opportunity.

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

Question 22. Examine this odds comparison function:

def find_best_odds(odds_by_book: dict[str, int]) -> tuple[str, int, float]:
    """Find the best available American odds from multiple books."""
    best_book = ""
    best_odds = -9999
    for book, odds in odds_by_book.items():
        if odds > best_odds:
            best_odds = odds
            best_book = book
    implied = abs(best_odds) / (abs(best_odds) + 100) if best_odds < 0 else 100 / (best_odds + 100)
    return best_book, best_odds, round(implied, 4)

(a) Trace through find_best_odds({"A": -110, "B": -105, "C": -115}).

(b) Identify a bug: what happens with find_best_odds({"A": +130, "B": -110})? Explain why the comparison odds > best_odds does not correctly identify the best odds for the bettor in all cases.

(c) Write a corrected version that handles both positive and negative American odds correctly.

def find_best_odds(odds_by_book: dict[str, int]) -> tuple[str, int, float]:
    if not odds_by_book:
        raise ValueError("No odds provided")
    best_book = max(odds_by_book, key=odds_by_book.get)
    best_odds = odds_by_book[best_book]
    if best_odds > 0:
        implied = 100 / (best_odds + 100)
    else:
        implied = abs(best_odds) / (abs(best_odds) + 100)
    return best_book, best_odds, round(implied, 4)

Question 23. Examine this line shopping value calculator:

def shopping_value(odds_used: int, worst_odds: int, best_odds: int, stake: float) -> dict:
    def to_decimal(american: int) -> float:
        return (american / 100) + 1 if american > 0 else (100 / abs(american)) + 1

    dec_used = to_decimal(odds_used)
    dec_worst = to_decimal(worst_odds)
    dec_best = to_decimal(best_odds)

    profit_used = stake * (dec_used - 1)
    profit_worst = stake * (dec_worst - 1)
    profit_best = stake * (dec_best - 1)

    savings_vs_worst = profit_used - profit_worst
    missed_vs_best = profit_best - profit_used
    efficiency = (profit_used - profit_worst) / (profit_best - profit_worst) if profit_best != profit_worst else 1.0

    return {
        "profit_if_win_used": round(profit_used, 2),
        "savings_vs_worst": round(savings_vs_worst, 2),
        "missed_vs_best": round(missed_vs_best, 2),
        "efficiency": round(efficiency, 4),
    }

(a) Trace shopping_value(-108, -115, -105, 100).

(b) What does the efficiency metric represent and what is a good target value?

(c) This function only calculates value when the bet wins. How would you modify it to calculate the expected shopping value incorporating win probability?

expected_savings = savings_vs_worst * win_prob
expected_missed = missed_vs_best * win_prob

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

Question 24. You maintain accounts at 5 sportsbooks and want to bet on 3 NFL games this Sunday. The available odds are:

(a) (2 points) For each game, identify the best book and calculate the implied probability savings versus the worst book.

(b) (2 points) You have $2,000 in each account. If you want to bet $500 on each game using the best odds, can you execute all three bets? If not, what is the constraint?

(c) (2 points) Calculate the total expected value of these three bets assuming your model gives KC a 54% cover probability, the Over a 55% chance, and BUF a 42% win probability. Use the best available odds.

(d) (2 points) If you could only use ONE sportsbook for all three bets, which would you choose? Calculate the total EV at each book and identify the optimal single-book choice.

Question 25. A bettor tracks their line shopping performance over 200 bets and provides this data:

(a) (2 points) Calculate the bettor's line shopping efficiency.

(b) (2 points) Estimate the dollar value of line shopping over these 200 bets compared to always using the worst available odds.

(c) (2 points) The bettor has positive ROI. Decompose their edge into (i) the portion from their model/picking ability and (ii) the portion from line shopping. Show your calculations.

(d) (2 points) If the bettor could improve their shopping efficiency from 79% to 95%, estimate the additional annual profit (assuming 600 bets per year at $200 each).

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