Chapter 1 Quiz: Introduction to Sports Betting
Instructions: Answer all 25 questions. This quiz is worth 100 points. You have 60 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 the "vigorish" (vig) in sports betting?
(A) The total amount of money wagered on a game by the public
(B) The commission charged by the sportsbook, built into the odds
(C) The profit earned by a bettor on a winning wager
(D) The difference between the opening and closing line
Answer
**(B) The commission charged by the sportsbook, built into the odds.** The vigorish (also called "vig," "juice," or "the take") is the sportsbook's built-in commission. It is reflected in the odds offered to bettors. For example, a standard -110/-110 line on both sides of a point spread means a bettor must risk \$110 to win \$100. If the sportsbook receives equal action on both sides, it collects \$110 from the loser and pays \$100 to the winner, netting \$10 regardless of the outcome. This margin is the vig.Question 2. A bettor places a \$100 wager at American odds of +250. If the bet wins, what is the total payout (stake plus profit)?
(A) \$250
(B) \$350
(C) \$150
(D) \$275
Answer
**(B) \$350.** With positive American odds, the profit on a \$100 stake is calculated as: Profit = Stake x (Odds / 100) = \$100 x (250 / 100) = \$250. Total payout = Stake + Profit = \$100 + \$250 = **\$350**. Option (A) is the profit only, not the total payout. Option (C) confuses the calculation with a -250 line. Option (D) is a common arithmetic error.Question 3. Decimal odds of 1.80 are equivalent to which American odds?
(A) +180
(B) -125
(C) -180
(D) +125
Answer
**(B) -125.** When decimal odds are less than 2.00, the equivalent American odds are negative. The conversion formula is: American = -100 / (Decimal - 1) = -100 / (1.80 - 1) = -100 / 0.80 = **-125**. Option (A) incorrectly treats 1.80 as 180. Option (C) simply negates 180. Option (D) applies the wrong conversion direction.Question 4. What is the implied probability of a bet offered at American odds of -200?
(A) 50.0%
(B) 60.0%
(C) 66.7%
(D) 75.0%
Answer
**(C) 66.7%.** For negative American odds, implied probability = |Odds| / (|Odds| + 100) = 200 / (200 + 100) = 200 / 300 = 0.6667 = **66.7%**. This means the sportsbook's odds imply a 66.7% chance of the favored outcome occurring. Note that this includes the vig, so the true probability may be lower.Question 5. A parlay bet requires:
(A) Betting on only one outcome but at adjusted odds
(B) All selected outcomes to win for the bet to pay out
(C) At least one of the selected outcomes to win for the bet to pay out
(D) The bettor to choose the correct score of the game
Answer
**(B) All selected outcomes to win for the bet to pay out.** A parlay (also called an accumulator or "acca" in the UK) combines multiple individual bets into a single wager. Every leg of the parlay must win for the bet to pay out. If any single leg loses, the entire parlay is lost. This all-or-nothing structure is what creates the higher potential payouts but also significantly lower probability of winning compared to individual straight bets.Question 6. Which of the following represents the LEAST favorable odds for a bettor backing the favorite?
(A) -130
(B) -145
(C) -160
(D) -115
Answer
**(C) -160.** For negative American odds, a larger absolute number is less favorable for the bettor because they must risk more to win the same amount. At -160, a bettor must risk \$160 to win \$100 (decimal odds 1.625). At -115, they only risk \$115 to win \$100 (decimal odds 1.870). The ranking from least to most favorable for the bettor is: -160, -145, -130, -115.Question 7. The 2018 U.S. Supreme Court ruling in Murphy v. NCAA resulted in:
(A) Legalizing sports betting nationwide under federal regulation
(B) Striking down the Professional and Amateur Sports Protection Act (PASPA), allowing states to legalize sports betting individually
(C) Imposing a federal tax on all sports betting revenue
(D) Banning daily fantasy sports nationwide
Answer
**(B) Striking down the Professional and Amateur Sports Protection Act (PASPA), allowing states to legalize sports betting individually.** The Supreme Court ruled 6-3 that PASPA (1992) violated the Tenth Amendment's anti-commandeering doctrine by prohibiting states from authorizing sports betting. The ruling did not legalize sports betting nationally; rather, it removed the federal prohibition and allowed each state to decide whether to legalize and regulate sports betting within its borders. This led to a wave of state-level legalization beginning in 2018.Question 8. In a point spread bet, "covering the spread" means:
(A) The favored team wins the game by any margin
(B) The underdog team wins the game outright
(C) The selected team's result, adjusted by the spread, is favorable
(D) The game total falls within the predicted range
Answer
**(C) The selected team's result, adjusted by the spread, is favorable.** Covering the spread means the team you bet on performed well enough relative to the point spread. For a favorite at -7, covering means winning by more than 7 points. For an underdog at +7, covering means losing by fewer than 7 points or winning outright. Option (A) is incorrect because the favorite could win but not cover (e.g., win by only 3 when favored by 7). Option (B) describes only one scenario in which an underdog covers.Question 9. Which of the following odds formats is predominantly used in the United Kingdom?
(A) American (moneyline)
(B) Decimal (European)
(C) Fractional (traditional)
(D) Hong Kong
Answer
**(C) Fractional (traditional).** Fractional odds (e.g., 5/1, 7/2, 11/4) are the traditional format used in the United Kingdom and Ireland, particularly at racetracks and in traditional high-street bookmakers. Decimal odds are more common in continental Europe, Australia, and on betting exchanges. American (moneyline) odds are the standard in the United States. Hong Kong odds are similar to decimal odds but represent only the profit (not including the returned stake).Question 10. A "sharp" bettor is best characterized as:
(A) A high-volume recreational bettor who wagers large amounts on favorites
(B) A professional or highly skilled bettor whose action often moves lines
(C) A sportsbook employee who sets the opening lines
(D) A bettor who exclusively places parlay wagers
Answer
**(B) A professional or highly skilled bettor whose action often moves lines.** "Sharp" refers to a sophisticated, typically professional bettor who uses quantitative methods, has a proven track record, and whose wagers are respected by sportsbooks. Sharp action often causes sportsbooks to move their lines. This is in contrast to "square" or recreational bettors, whose wagers are less likely to move lines. Sportsbooks track sharp bettors and may limit their accounts over 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: In a standard -110/-110 market, the sportsbook's vigorish is approximately 4.55%.
Answer
**True.** Each side at -110 has an implied probability of 110/210 = 52.38%. The overround is 52.38% + 52.38% = 104.76%. The vig percentage is (1 - 1/1.0476) x 100 = (1 - 0.9545) x 100 = **4.55%**. This means that on average, for every \$100 wagered across both sides, the sportsbook retains approximately \$4.55.Question 12. True or False: Decimal odds of 2.00 imply a 50% probability, and the equivalent American odds are +100.
Answer
**True.** Decimal odds of 2.00 mean a \$1 bet returns \$2 (the original stake plus \$1 profit). The implied probability is 1/2.00 = 0.50 = 50%. The American odds equivalent: since decimal 2.00 means even money, American odds = +100. A \$100 bet at +100 wins \$100 in profit, for a total payout of \$200, identical to a \$100 bet at decimal 2.00.Question 13. True or False: A parlay is always a negative expected value bet, even if the individual legs are each positive expected value.
Answer
**False.** If each individual leg of a parlay has positive expected value (+EV), the parlay itself will also have positive expected value. The expected value of a parlay is the product of the individual legs' expected returns minus the stake. If each leg has a positive edge, combining them into a parlay amplifies the potential payout while maintaining a positive overall expectation. Parlays are negative EV for most bettors because the standard vig-inflated odds make individual legs negative EV, and the parlay compounds those disadvantages. But if the legs are genuinely +EV, the parlay is too.Question 14. True or False: The closing line of a sports betting market is generally considered a more accurate predictor of game outcomes than the opening line.
Answer
**True.** Research consistently shows that closing lines incorporate more information than opening lines. Between open and close, sharp bettors, injury reports, weather updates, and public action all contribute to price discovery. The closing line represents the market's final, most-informed estimate of the true probabilities. Consistently beating the closing line (i.e., getting a better price at the time of your bet than the final line) is widely considered the best indicator of long-term betting skill.Question 15. True or False: In the United States, sports betting winnings are not subject to federal income tax if the bettor's annual winnings are under \$600.
Answer
**False.** All gambling winnings are subject to federal income tax regardless of the amount. The \$600 threshold (or 300:1 odds for certain wagers) is the reporting threshold at which sportsbooks are required to issue a W-2G form and may withhold taxes. However, the tax obligation exists on all gambling income. Bettors are legally required to report all winnings on their tax returns. The \$600 figure is a reporting trigger for the sportsbook, not a tax exemption threshold for the bettor.Section 3: Fill in the Blank (3 questions, 4 points each = 12 points)
Question 16. To convert negative American odds to decimal odds, use the formula:
Decimal Odds = 1 + (100 / __________)
Answer
**|American Odds|** (the absolute value of the American odds) The full formula is: Decimal = 1 + (100 / |American|). For example, American odds of -150: Decimal = 1 + (100/150) = 1 + 0.6667 = 1.6667. This works because negative American odds tell you how much you must wager to win \$100, so dividing 100 by that amount gives the profit per dollar wagered, and adding 1 accounts for the return of the original stake.Question 17. The total of all implied probabilities derived from the odds in a betting market is called the __________. When this value exceeds 100%, the excess represents the sportsbook's built-in margin.
Answer
**Overround** (also accepted: "vig margin," "book percentage," or "total implied probability") The overround is the sum of implied probabilities across all outcomes in a market. In a fair (vig-free) market, it would equal exactly 100%. In practice, it always exceeds 100%. For example, a -110/-110 market has an overround of 104.76%. The overround minus 100% indicates the sportsbook's theoretical edge. A lower overround is more favorable for bettors.Question 18. A betting market where informed participants quickly correct mispricings, making it difficult for any bettor to consistently profit, is described as __________ efficient.
Answer
**Market** efficient (or specifically, **semi-strong form** efficient, or simply **efficient**) This concept parallels the Efficient Market Hypothesis (EMH) from financial economics. In a market-efficient sports betting environment, odds rapidly incorporate all publicly available information, making it difficult for bettors to find consistent edges. Sports betting markets are generally considered semi-strong form efficient, meaning public information is quickly priced in, but private information (e.g., non-public injury details) may still provide an edge.Section 4: Short Answer (3 questions, 5 points each = 15 points)
Answer each question in 3-5 sentences.
Question 19. Explain the difference between a "moneyline" bet and a "point spread" bet. Under what circumstances might a bettor prefer one over the other?
Answer
A **moneyline** bet is a wager on which team will win the game outright, with no point spread adjustment. The odds reflect the perceived difference in team strength: favorites have negative odds (risk more to win less) and underdogs have positive odds (risk less to win more). A **point spread** bet evens the playing field by adding or subtracting points from a team's final score; the bettor is wagering on the margin of victory rather than simply who wins. A bettor might prefer the moneyline when they believe an underdog has a good chance of winning outright (moneyline underdogs offer larger payouts than spread underdogs), or when the spread is a key number (like 3 or 7 in football) and they want to avoid the risk of a push or bad beat. A bettor might prefer the spread when backing a heavy favorite, since the moneyline payout on large favorites is very small relative to the risk (e.g., -300 requires risking \$300 to win \$100).Question 20. Why is expected value (EV) a more important metric than win rate for evaluating a bettor's long-term profitability? Provide a brief numerical example to support your explanation.
Answer
**Expected value** measures the average profit or loss per bet over the long run, accounting for both the probability of winning and the payout. Win rate alone is insufficient because it ignores the odds and payout structure. A bettor could have a high win rate but still lose money if they consistently bet heavy favorites at poor odds, and conversely, a bettor could have a low win rate but be profitable by targeting underdogs at valuable prices. **Example:** Bettor A wins 60% of bets at -200 (risk \$200 to win \$100). EV per bet = (0.60 x \$100) - (0.40 x \$200) = \$60 - \$80 = **-\$20** (losing \$20 per bet on average despite a 60% win rate). Bettor B wins only 35% of bets at +300 (risk \$100 to win \$300). EV per bet = (0.35 x \$300) - (0.65 x \$100) = \$105 - \$65 = **+\$40** (profiting \$40 per bet on average with only a 35% win rate). Bettor B has a superior long-term strategy despite a much lower win rate.Question 21. Describe two key differences between a traditional sportsbook model and a betting exchange model (such as Betfair). What advantages does the exchange model offer to sophisticated bettors?
Answer
**Difference 1: Counterparty.** In a traditional sportsbook model, the bettor wagers against the house (the sportsbook), which sets the odds and manages risk. In an exchange model, bettors wager against each other, with the exchange acting as a neutral intermediary that matches buy (back) and sell (lay) orders. **Difference 2: Pricing and margin.** A sportsbook builds its profit margin (vig) directly into the odds. An exchange charges a commission (typically 2-5%) on net winnings, resulting in significantly lower effective margins. The odds on an exchange are set by market participants, often resulting in prices closer to true probabilities. **Advantages for sophisticated bettors:** Exchanges allow bettors to "lay" outcomes (bet against something happening), which is not possible at a traditional sportsbook. The lower margins mean less of the bettor's edge is consumed by the vig. Additionally, exchanges generally do not limit or ban winning bettors, since profitable players contribute commission revenue rather than costing the platform money.Section 5: Code Analysis (2 questions, 6 points each = 12 points)
Read the provided Python code and answer the questions. You do not need to execute the code.
Question 22. Examine the following Python function:
def convert_odds(american_odds):
if american_odds > 0:
decimal = (american_odds / 100) + 1
implied_prob = 100 / (american_odds + 100)
else:
decimal = (100 / american_odds) + 1
implied_prob = american_odds / (american_odds - 100)
return round(decimal, 4), round(implied_prob, 4)
(a) This function contains a bug that will produce incorrect results for negative American odds. Identify the bug and explain what the incorrect output would be for an input of -150.
(b) Write the corrected version of the problematic lines.
Answer
**(a)** The bug is in the `else` block (negative American odds). Both calculations use the raw negative value instead of the absolute value, producing wrong results. For `american_odds = -150`: - `decimal = (100 / -150) + 1 = -0.6667 + 1 = 0.3333` — This is incorrect. Decimal odds should always be greater than 1.0. The correct value is `1 + (100 / 150) = 1.6667`. - `implied_prob = -150 / (-150 - 100) = -150 / -250 = 0.6` — This line actually produces the correct numerical result (0.60 = 60%) by coincidence of the double negative, but the logic is not transparent. A more robust formulation uses absolute values. The decimal odds calculation is clearly broken for negative inputs. **(b)** Corrected version:def convert_odds(american_odds):
if american_odds > 0:
decimal = (american_odds / 100) + 1
implied_prob = 100 / (american_odds + 100)
else:
decimal = (100 / abs(american_odds)) + 1
implied_prob = abs(american_odds) / (abs(american_odds) + 100)
return round(decimal, 4), round(implied_prob, 4)
Question 23. Examine the following Python function that calculates the vigorish:
def calculate_vig(odds_a, odds_b):
"""Calculate vig given American odds for both sides of a two-way market."""
def implied_prob(american):
if american > 0:
return 100 / (american + 100)
else:
return abs(american) / (abs(american) + 100)
prob_a = implied_prob(odds_a)
prob_b = implied_prob(odds_b)
overround = prob_a + prob_b
vig = (1 - (1 / overround)) * 100
fair_a = prob_a / overround
fair_b = prob_b / overround
return {
"implied_a": round(prob_a, 4),
"implied_b": round(prob_b, 4),
"overround": round(overround, 4),
"vig_pct": round(vig, 2),
"fair_prob_a": round(fair_a, 4),
"fair_prob_b": round(fair_b, 4),
}
(a) Trace through the execution of calculate_vig(-110, -110) and write the exact return value.
(b) Now trace calculate_vig(-150, +130) and write the exact return value.
(c) Explain in one sentence what fair_a and fair_b represent conceptually.
Answer
**(a)** `calculate_vig(-110, -110)`: - `prob_a = 110 / (110 + 100) = 110 / 210 = 0.52381 -> round to 0.5238` - `prob_b = 110 / (110 + 100) = 110 / 210 = 0.52381 -> round to 0.5238` - `overround = 0.52381 + 0.52381 = 1.04762 -> round to 1.0476` - `vig = (1 - 1/1.04762) * 100 = (1 - 0.95455) * 100 = 4.545 -> round to 4.55` - `fair_a = 0.52381 / 1.04762 = 0.5 -> round to 0.5` - `fair_b = 0.52381 / 1.04762 = 0.5 -> round to 0.5` Return value:{"implied_a": 0.5238, "implied_b": 0.5238, "overround": 1.0476, "vig_pct": 4.55, "fair_prob_a": 0.5, "fair_prob_b": 0.5}
{"implied_a": 0.6, "implied_b": 0.4348, "overround": 1.0348, "vig_pct": 3.36, "fair_prob_a": 0.5798, "fair_prob_b": 0.4202}
Section 6: Applied Problems (2 questions, 8 points each = 16 points)
Question 24. You are evaluating a bet on an upcoming NBA game. The sportsbook offers the following:
| Bet | Odds (American) |
|---|---|
| Milwaukee Bucks moneyline | -180 |
| Indiana Pacers moneyline | +155 |
| Bucks -4.5 | -110 |
| Pacers +4.5 | -110 |
| Over 228.5 | -105 |
| Under 228.5 | -115 |
Your model outputs the following estimates:
- Bucks win probability: 62%
- Bucks win by 5+ points: 45%
- Over 228.5 probability: 55%
(a) (2 points) Calculate the implied probability for the Bucks moneyline at -180 and the expected value of a \$100 bet using your model's win probability.
(b) (2 points) Calculate the expected value of a \$100 bet on the Bucks -4.5 at -110, using your model's estimate that the Bucks win by 5+ points with probability 45%.
(c) (2 points) Calculate the expected value of a \$100 bet on the Over 228.5 at -105.
(d) (2 points) Based on your EV calculations, which bet(s) would you recommend and why? If none are positive EV, explain what odds you would need to make each bet worthwhile.
Answer
**(a)** Bucks moneyline at -180: - Implied probability = 180 / (180 + 100) = 180 / 280 = **64.29%** - Profit if win: \$100 x (100/180) = \$55.56 - EV = (0.62 x \$55.56) - (0.38 x \$100) = \$34.44 - \$38.00 = **-\$3.56** - This is a negative EV bet. Your model says 62% but the line implies 64.29%, meaning the sportsbook is pricing the Bucks as more likely to win than your model estimates. **(b)** Bucks -4.5 at -110: - Implied probability = 110 / (110 + 100) = 110 / 210 = 52.38% - Profit if win: \$100 x (100/110) = \$90.91 - EV = (0.45 x \$90.91) - (0.55 x \$100) = \$40.91 - \$55.00 = **-\$14.09** - This is a negative EV bet. Your model's 45% probability falls well below the implied probability threshold of 52.38%. **(c)** Over 228.5 at -105: - Implied probability = 105 / (105 + 100) = 105 / 205 = 51.22% - Profit if win: \$100 x (100/105) = \$95.24 - EV = (0.55 x \$95.24) - (0.45 x \$100) = \$52.38 - \$45.00 = **+\$7.38** - This is a **positive EV bet**. Your model's 55% probability exceeds the implied probability of 51.22%. **(d)** **Recommendation:** The only positive EV bet is the **Over 228.5 at -105**, with an EV of +\$7.38 per \$100 wagered (+7.38% ROI). This is the recommended bet. The Bucks moneyline would require odds of approximately -163 or better (implied probability <= 62%) to become +EV. The Bucks -4.5 would require approximately +122 or better (implied probability <= 45%) to become +EV. These thresholds can be derived by setting EV = 0 and solving for the payout.Question 25. A friend shows you a promotional offer from a new sportsbook: "Bet \$500 risk-free on any sporting event! If your bet loses, we will refund your \$500 as a free bet (free bet winnings do not include the stake)."
Your friend plans to use this on a -110 moneyline bet. Analyze this offer completely:
(a) (2 points) If the initial \$500 bet wins at -110, what is the profit? What is the profit if it loses and the \$500 free bet (at -110) subsequently wins? What if both the original bet and the free bet lose?
(b) (2 points) Calculate the expected value of this promotion assuming -110 odds on both the initial bet and the free bet, and a 50% true win probability for each.
(c) (2 points) Your friend asks: "Should I use the free bet on a heavy favorite at -300 or a big underdog at +300?" Advise them by calculating the expected free bet value at each of those odds (assume true probabilities match implied probabilities with vig removed).
(d) (2 points) What is the optimal strategy for maximizing the value of this risk-free bet promotion? Consider both the initial bet and the free bet strategy.
Answer
**(a)** Three scenarios: 1. **Initial bet wins at -110:** Profit = \$500 x (100/110) = **\$454.55**. No free bet needed. 2. **Initial bet loses, free bet wins at -110:** The initial bet loses \$500. The free bet returns only the winnings (not the \$500 stake): \$500 x (100/110) = \$454.55. Net result: -\$500 + \$454.55 = **-\$45.45** (small net loss). 3. **Both bets lose:** Net result: -\$500 + \$0 = **-\$500** (full loss of original stake; free bet returns nothing). **(b)** Expected value with 50% true win probability at -110 for both bets: - P(initial wins) = 0.50 -> profit \$454.55 - P(initial loses, free bet wins) = 0.50 x 0.50 = 0.25 -> net -\$45.45 - P(initial loses, free bet loses) = 0.50 x 0.50 = 0.25 -> net -\$500 EV = (0.50 x \$454.55) + (0.25 x -\$45.45) + (0.25 x -\$500) EV = \$227.28 - \$11.36 - \$125.00 = **+\$90.91** The promotion has a positive expected value of approximately **\$90.91**, representing roughly an 18.2% return on the \$500 initial bet. **(c)** The free bet value depends on odds. Since the free bet does not return the stake, higher odds (bigger underdogs) yield more value. **Free bet at -300** (vig-removed true prob approximately 73.2%): - Winnings if it hits: \$500 x (100/300) = \$166.67 - Expected free bet value = 0.732 x \$166.67 = **\$122.00** **Free bet at +300** (vig-removed true prob approximately 26.8%): - Winnings if it hits: \$500 x (300/100) = \$1,500 - Expected free bet value = 0.268 x \$1,500 = **\$402.00** The +300 underdog free bet has an expected value of \$402.00, more than triple the -300 favorite's \$122.00. **The underdog is the far better choice for the free bet** because the free bet's stake is not returned in either case, so you want to maximize the potential payout per dollar of free bet "risk." **(d)** Optimal strategy for maximizing the promotion value: 1. **Initial bet:** Place the \$500 initial bet on a **heavy favorite** (large negative odds) to maximize the probability that it wins outright, since a win returns both the stake and the profit. The opportunity cost of "using up" the free bet is minimized when the initial bet wins. At -300, there is roughly a 73% chance the initial bet wins, yielding \$166.67 profit without needing the free bet. 2. **Free bet (if triggered):** Use the free bet on a **large underdog** (high positive odds) to maximize the expected value of the free bet. Since the free bet stake is not returned, the breakeven point and optimal play shifts heavily toward longshots. The ideal target is typically in the +200 to +500 range, balancing payout size with a reasonable chance of hitting. 3. **Hedge consideration:** An advanced approach is to hedge: place the initial \$500 on one side at one sportsbook, and simultaneously bet the other side at another sportsbook. This can guarantee triggering the free bet while limiting the net cost, then use the free bet on a longshot for maximum expected value. The exact optimal hedge depends on the available odds at both books.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% |