Chapter 7 Quiz: Probability Distributions in Betting

Instructions: Select the best answer for each question. Answers are hidden at the end of the document.

Question 1

Which probability distribution is most commonly used to model the number of goals scored by a single team in a soccer match?

A) Normal distribution B) Binomial distribution C) Poisson distribution D) Beta distribution

Question 2

A Poisson distribution with lambda = 1.5 models goals scored per match. What is the probability of exactly 2 goals?

A) 0.2231 B) 0.2510 C) 0.3347 D) 0.1255

Question 3

Which property distinguishes the Poisson distribution from the Binomial distribution in modeling scoring events?

A) The Poisson distribution requires a fixed number of trials B) The Poisson distribution models the number of events in a fixed interval without a defined number of trials C) The Poisson distribution only applies to continuous data D) The Poisson distribution assumes events are dependent

Question 4

An NBA team has a per-game win probability of 0.60 across an 82-game season. What is the standard deviation of their win total under the binomial model?

A) 4.43 B) 6.27 C) 19.68 D) 49.20

Question 5

In the context of NFL point spread analysis, what does the standard deviation of approximately 13.86 points represent?

A) The average point spread set by bookmakers B) The typical variability of actual game margins around the spread line C) The maximum expected margin of victory D) The bookmaker's profit margin on spread bets

Question 6

The Beta distribution is uniquely suited for Bayesian modeling of win probabilities because:

A) It is the only distribution defined on the interval [0, 1] B) It is the conjugate prior to the Binomial likelihood, making posterior calculations analytically tractable C) It always produces symmetric distributions D) It requires no prior information

Question 7

Using a Poisson model with home lambda = 1.4 and away lambda = 1.0 (assuming independence), what is the probability of a 0-0 draw?

A) P(0) x P(0) = e^(-1.4) x e^(-1.0) = e^(-2.4) approximately 0.0907 B) 0.2466 x 0.3679 approximately 0.1353 C) 0.50 x 0.50 = 0.25 D) Cannot be calculated without additional information

Question 8

If NFL spread margins follow a Normal(0, 13.86) distribution, what is the approximate probability of a 7-point favorite covering the spread?

A) 0.50 B) 0.31 C) 0.69 D) 0.42

Question 9

A bettor starts with a Beta(1, 1) prior for a team's win probability and observes 9 wins in 15 games. What is the posterior distribution?

A) Beta(9, 15) B) Beta(10, 7) C) Beta(9, 6) D) Beta(10, 16)

Question 10

Which of the following is an assumption of the Poisson distribution that may be violated in sports scoring?

A) Events must occur in whole numbers B) Events occur independently of each other C) The distribution must be symmetric D) The sample size must be greater than 30

Question 11

The Negative Binomial distribution is preferred over the Poisson when the data exhibits:

A) Underdispersion (variance less than mean) B) Equidispersion (variance equals mean) C) Overdispersion (variance greater than mean) D) Zero inflation (excess zeros)

Question 12

An NFL team has a normal model for game margins with mean = +3.5 (home favorite) and standard deviation = 13.86. What is the probability they win outright (margin > 0)?

A) 0.40 B) 0.50 C) 0.60 D) 0.75

Question 13

In the context of the Over/Under market for soccer, which calculation gives P(Over 2.5 goals) when total goals follow Poisson(lambda)?

A) 1 - P(X = 0) - P(X = 1) B) 1 - P(X = 0) - P(X = 1) - P(X = 2) C) P(X = 3) + P(X = 4) + P(X = 5) D) Both B and C are equivalent

Question 14

The Central Limit Theorem is relevant to sports betting because:

A) It proves that all sports outcomes follow a normal distribution B) It explains why the sum or average of many independent random events tends toward a normal distribution, justifying normal models for aggregate scores C) It guarantees that betting strategies will converge to expected value D) It only applies to continuous distributions

Question 15

A Beta(50, 50) distribution represents which of the following beliefs about a team's win probability?

A) High confidence that the win probability is near 0.50 B) Complete uncertainty about the win probability C) High confidence that the win probability is near 0.75 D) A belief that the team always wins

Question 16

When using a Poisson model for soccer, the assumption of independence between the two teams' goal-scoring processes:

A) Is always perfectly satisfied in practice B) Is a simplification; the Dixon-Coles model introduces a correction for low-scoring dependencies C) Means the model cannot be used for betting D) Requires that both teams have the same scoring rate

Question 17

A binomial distribution with n = 17 and p = 0.65 models an NFL team's season wins. What is the expected number of wins?

A) 8.50 B) 11.05 C) 12.00 D) 13.60

Question 18

The variance of a Poisson distribution with parameter lambda = 2.3 is:

A) 1.52 B) 2.30 C) 5.29 D) 4.60

Question 19

Which scenario best illustrates the use of the Beta distribution in sports betting?

A) Predicting the exact number of goals in a soccer match B) Estimating the posterior probability that a pitcher's true strikeout rate exceeds 25% C) Calculating the probability of covering a point spread D) Modeling the total number of wins in a season

Question 20

A sportsbook offers decimal odds of 3.20 on a draw. This implies a probability of approximately:

A) 0.3125 B) 0.3200 C) 0.6875 D) 0.2200

Question 21

If you model point spread margins as Normal(0, sigma) and observe that your model predicts a 54% cover rate but the sportsbook requires 52.38% to break even at -110, the edge is:

A) 1.62 percentage points B) 3.08 percentage points C) 54.00 percentage points D) Cannot be determined without knowing sigma

Question 22

The Poisson distribution P(X = k) = (lambda^k * e^(-lambda)) / k! requires which input parameters?

A) Mean and standard deviation B) Number of trials and success probability C) Only the rate parameter lambda D) Alpha and beta shape parameters

Question 23

When fitting distributions to historical NFL game totals, which goodness-of-fit metric penalizes model complexity to prevent overfitting?

A) Chi-squared test statistic B) Log-likelihood C) AIC (Akaike Information Criterion) D) Mean absolute error

Question 24

A bettor with a Beta(15, 10) posterior believes a team's true win rate is approximately 60%. They see market odds implying 50%. The probability that the team's true rate exceeds 50% can be found by:

A) Calculating the CDF of Beta(15, 10) at 0.50 and subtracting from 1 B) Taking the posterior mean and comparing to 0.50 C) Performing a t-test D) Using the Poisson CDF

Question 25

The "law of small numbers" fallacy in sports betting refers to:

A) The mathematical proof that small samples are always representative B) The mistaken belief that small samples should closely mirror the true underlying probabilities, leading bettors to overreact to short-term results C) The principle that small bets always have positive expected value D) The rule that you should never bet on outcomes with probability less than 10%

Answer Key

End of Chapter 7 Quiz