Chapter 3 Quiz: Probability Fundamentals
Instructions: Answer all 25 questions. For multiple choice, select the single best answer. For true/false, provide a brief justification. For fill-in-the-blank, provide the exact formula or term. For short answer and code analysis, provide concise but complete responses.
Multiple Choice (10 Questions)
Question 1
A prediction market offers contracts on which of four candidates will win an election. The prices are: A = $0.40, B = $0.30, C = $0.20, D = $0.15. What can you conclude?
- (a) The market is perfectly efficient
- (b) The prices represent a valid probability distribution
- (c) The prices sum to more than $1, indicating overround or transaction costs
- (d) Candidate A will definitely win
Answer
**(c)** The prices sum to $0.40 + $0.30 + $0.20 + $0.15 = $1.05, which exceeds $1.00. In a valid probability distribution, probabilities must sum to exactly 1. The excess ($0.05) represents the market's overround, which covers transaction costs and provides profit for the market maker. To derive implied probabilities, you would divide each price by 1.05.Question 2
You buy a binary contract at $0.35 and believe the true probability of the event is 0.50. What is your expected value per contract?
- (a) $0.35
- (b) $0.50
- (c) $0.15
- (d) -$0.15
Answer
**(c)** For a binary contract, EV = q - p = 0.50 - 0.35 = $0.15. Your estimated probability (0.50) exceeds the price you pay (0.35), so you have a positive expected value of $0.15 per contract.Question 3
Bayes' theorem states that P(A|B) = P(B|A) * P(A) / P(B). In this formula, P(A) is called:
- (a) The posterior probability
- (b) The likelihood
- (c) The prior probability
- (d) The evidence
Answer
**(c)** P(A) is the **prior probability** --- your belief about A before observing evidence B. P(B|A) is the likelihood, P(B) is the evidence (or marginal likelihood), and P(A|B) is the posterior probability.Question 4
If P(A) = 0.3, P(B) = 0.5, and A and B are independent, what is P(A and B)?
- (a) 0.8
- (b) 0.15
- (c) 0.2
- (d) 0.65
Answer
**(b)** For independent events, P(A and B) = P(A) * P(B) = 0.3 * 0.5 = 0.15. Independence means knowing one event tells you nothing about the other, so the joint probability is simply the product of the marginals.Question 5
A binary contract is priced at $0.50. What is the variance of its payout (which is either $0 or $1)?
- (a) 0.50
- (b) 0.25
- (c) 0.125
- (d) 0.75
Answer
**(b)** For a Bernoulli random variable with parameter p, Var(X) = p(1-p) = 0.50 * 0.50 = 0.25. This is the maximum possible variance for a Bernoulli variable, which makes sense because p = 0.50 represents maximum uncertainty.Question 6
You start with a Beta(3, 7) prior for the probability of an event. You then observe the event occur 4 times and not occur 2 times. What is the posterior distribution?
- (a) Beta(4, 2)
- (b) Beta(7, 9)
- (c) Beta(3.4, 7.2)
- (d) Beta(12, 14)
Answer
**(b)** With the Beta-Binomial conjugate model, the update rule is: Beta(alpha + successes, beta + failures) = Beta(3 + 4, 7 + 2) = Beta(7, 9). The posterior mean is 7/(7+9) = 0.4375, up from the prior mean of 3/(3+7) = 0.30.Question 7
The Law of Large Numbers tells us that:
- (a) After a losing streak, a winning streak is more likely
- (b) The sample average converges to the expected value as the number of trials increases
- (c) All random variables eventually become normally distributed
- (d) Larger bets are more likely to be profitable
Answer
**(b)** The LLN states that the sample average converges to the population mean (expected value) as the number of independent trials increases. It does NOT say that losses will be compensated by future wins (that would be the gambler's fallacy). Option (c) describes the Central Limit Theorem (and even that is about the distribution of the average, not individual variables).Question 8
In the odds form of Bayes' theorem, if the prior odds are 1:3 and the likelihood ratio is 6, what are the posterior odds?
- (a) 2:1
- (b) 6:3
- (c) 1:18
- (d) 7:3
Answer
**(a)** Posterior odds = Likelihood ratio * Prior odds = 6 * (1/3) = 2. So the posterior odds are 2:1. Converting to probability: 2/(2+1) = 2/3 = 0.667. The evidence was strong enough to shift the belief from 25% (1:3 odds) to 66.7% (2:1 odds).Question 9
Which probability distribution is the conjugate prior for the Binomial likelihood?
- (a) Normal
- (b) Poisson
- (c) Beta
- (d) Gamma
Answer
**(c)** The Beta distribution is the conjugate prior for the Binomial (and Bernoulli) likelihood. This means that if the prior on the probability parameter p is Beta(alpha, beta), and we observe data from a Binomial distribution, the posterior is also a Beta distribution with updated parameters: Beta(alpha + successes, beta + failures).Question 10
You want to calculate P(A or B) where P(A) = 0.40, P(B) = 0.50, and P(A and B) = 0.25. The answer is:
- (a) 0.90
- (b) 0.65
- (c) 0.40
- (d) 0.15
Answer
**(b)** Using the addition rule: P(A or B) = P(A) + P(B) - P(A and B) = 0.40 + 0.50 - 0.25 = 0.65. We subtract the intersection to avoid double-counting outcomes that are in both A and B.True/False (5 Questions)
Question 11
True or False: If two events are mutually exclusive, they must be independent.
Answer
**False.** Mutually exclusive events are actually the opposite of independent (assuming both have nonzero probability). If A and B are mutually exclusive, then P(A and B) = 0. But for independence, we need P(A and B) = P(A) * P(B), which would only be 0 if one of the events had probability 0. Since knowing A occurred tells you B definitely did NOT occur, they are maximally dependent. For example, "Candidate A wins" and "Candidate B wins" are mutually exclusive and highly dependent.Question 12
True or False: If a prediction market contract is priced at $0.80, buying it always has negative expected value since you risk $0.80 to win only $0.20.
Answer
**False.** The expected value depends on the true probability, not just the potential payout. If the true probability is 0.90, then EV = 0.90 - 0.80 = $0.10, which is positive. The asymmetric payoff ($0.20 gain vs. $0.80 loss) is compensated by the asymmetric probabilities (90% chance of winning vs. 10% chance of losing). What matters is EV = q - p, not the raw payout amounts.Question 13
True or False: The Central Limit Theorem guarantees that the sum of any 30 or more random variables will be approximately normally distributed.
Answer
**False.** The CLT requires the random variables to be independent and identically distributed (or at least satisfy certain regularity conditions). The "n = 30" rule of thumb is a rough guideline, not a guarantee. For highly skewed distributions (e.g., Pareto distributions with infinite variance), the CLT may require much larger n or may not apply at all. Additionally, the CLT describes the limiting distribution --- the approximation quality at finite n depends on the shape of the underlying distribution.Question 14
True or False: When performing sequential Bayesian updating, the order in which you process independent pieces of evidence does not affect the final posterior.
Answer
**True.** If the evidence is conditionally independent given the hypothesis, the order of processing does not matter. In the odds form, the posterior odds equal the prior odds multiplied by each likelihood ratio. Since multiplication is commutative, the order is irrelevant. Mathematically: LR1 * LR2 * LR3 * prior_odds = LR3 * LR1 * LR2 * prior_odds. The final posterior is the same regardless of the sequence.Question 15
True or False: A Beta(100, 100) prior and a Beta(1, 1) prior will produce nearly identical posteriors after observing 10,000 data points.
Answer
**True.** With a very large amount of data (10,000 observations), the data overwhelms both priors. The Beta(100, 100) prior is equivalent to having 200 "pseudo-observations," which is negligible compared to 10,000 actual observations. Both posteriors will be very close to Beta(alpha + successes, beta + failures), where the alpha and beta contributions from the prior are dwarfed by the data. This illustrates the principle that with sufficient data, the choice of prior becomes irrelevant (Bayesian convergence / washing out of priors).Fill in the Blank (4 Questions)
Question 16
The complement rule states: P(A^c) = _____
Answer
**1 - P(A)** The complement rule follows directly from Kolmogorov's second axiom (P(S) = 1) and third axiom (additivity). Since A and A^c are mutually exclusive and exhaustive: P(A) + P(A^c) = P(S) = 1, therefore P(A^c) = 1 - P(A).Question 17
Bayes' theorem: P(A|B) = _____
Answer
**P(B|A) * P(A) / P(B)** This is derived by combining the two expressions for the joint probability: P(A and B) = P(A|B) * P(B) = P(B|A) * P(A). Solving for P(A|B) gives the formula.Question 18
For a binary prediction market contract purchased at price p with true probability q, the expected value is: EV = _____
Answer
**q - p** Derivation: EV = q * (1-p) + (1-q) * (-p) = q - qp - p + qp = q - p. The expected value is simply the difference between the true probability and the price paid.Question 19
The Beta-Binomial conjugate update rule: After observing s successes and f failures, Beta(alpha, beta) updates to _____
Answer
**Beta(alpha + s, beta + f)** This elegant update rule is why the Beta-Binomial pair is called "conjugate." The posterior has the same functional form (Beta) as the prior, with parameters adjusted by simply adding the observed counts. The posterior mean is (alpha + s) / (alpha + beta + s + f).Short Answer (3 Questions)
Question 20
Explain what the "likelihood ratio" is in Bayesian updating and why a likelihood ratio of exactly 1.0 means the evidence is uninformative.
Answer
The **likelihood ratio** (also called the Bayes factor) is the ratio of how probable the evidence is under the hypothesis versus under the alternative: LR = P(Evidence | Hypothesis) / P(Evidence | Not Hypothesis) It measures the **diagnostic strength** of the evidence --- how much the evidence distinguishes between the hypothesis being true and being false. A likelihood ratio of 1.0 means P(E|H) = P(E|not H): the evidence is equally likely whether the hypothesis is true or false. In the odds form of Bayes' theorem, posterior odds = LR * prior odds. When LR = 1, posterior odds = prior odds, meaning the evidence does not change your beliefs at all. The evidence is **uninformative** because observing it gives you no reason to update in either direction. For example, if you are trying to determine whether a coin is fair or biased, and your evidence is "the sun rose this morning," that evidence is equally likely whether the coin is fair or biased (LR = 1), so it provides no information about the coin.Question 21
A prediction market has a contract at $0.95 (95% implied probability). You believe the true probability is 0.97. Calculate the EV. Then explain why many experienced traders would still not take this trade despite the positive EV.
Answer
EV = q - p = 0.97 - 0.95 = $0.02 per contract. Despite the positive EV, experienced traders may avoid this trade for several reasons: 1. **Tiny edge, large risk:** You risk $0.95 to gain $0.05, a 19:1 risk-reward ratio. Even a small error in your probability estimate (e.g., true probability is actually 0.93 instead of 0.97) flips the EV negative, and your loss is much larger than your potential gain. 2. **Calibration uncertainty:** At extreme probabilities, small estimation errors lead to disproportionately large EV swings. Being wrong by 2 percentage points at p=0.50 is very different from being wrong by 2 percentage points at p=0.95. 3. **Capital inefficiency:** You tie up $0.95 per contract to earn an expected $0.02. The return on capital (2.1%) may be lower than alternatives. 4. **Tail risk:** The 3% chance of losing $0.95 can be devastating if you size the position aggressively. The Kelly criterion would recommend a very small position for such a thin edge. 5. **Cromwell's rule concern:** Very high certainty deserves scrutiny. Events thought to be "nearly certain" sometimes don't happen.Question 22
Describe the relationship between the Law of Large Numbers and the concept of "edge" in prediction market trading. Why does an edge only guarantee profitability in the long run, not the short run?
Answer
An **edge** in prediction market trading means your estimated probability q differs from the market price p in your favor (q > p for a buy, q < p for a short). This gives you a positive expected value per trade (EV = q - p > 0). The **Law of Large Numbers** connects edge to profitability by guaranteeing that as the number of independent trades grows, your average profit per trade converges to the true EV. This means: - **Long run:** If you consistently make positive EV trades, your average profit will eventually settle near your true edge. Profitability becomes near-certain with enough trades. - **Short run:** Individual trades are governed by randomness. Even with a 10% edge, you lose 45% of the time (if q = 0.55 and p = 0.45). Over 10 trades, the variance is large relative to the expected gain. It is entirely possible to lose money over any short period despite having a genuine edge. The mathematical reason is that the standard deviation of the average profit shrinks as 1/sqrt(n). After 10 trades, your average is noisy (std proportional to 1/sqrt(10) = 0.32). After 10,000 trades, the noise is much smaller (1/sqrt(10000) = 0.01), and the signal (your edge) dominates. This is why professional traders focus on bankroll management and position sizing --- they need to survive the short-term variance long enough for the LLN to work in their favor.Code Analysis (3 Questions)
Question 23
What does the following code compute? Identify the probability concept being implemented and explain the output.
def mystery(p_h, p_e_given_h, p_e_given_not_h):
p_not_h = 1 - p_h
p_e = p_e_given_h * p_h + p_e_given_not_h * p_not_h
return (p_e_given_h * p_h) / p_e
result = mystery(0.20, 0.90, 0.05)
print(f"Result: {result:.4f}")
Answer
This code implements **Bayes' theorem**. It computes P(H|E) --- the posterior probability of hypothesis H given evidence E. Breaking it down: - `p_h = 0.20` is the **prior** P(H) - `p_e_given_h = 0.90` is the **likelihood** P(E|H) - `p_e_given_not_h = 0.05` is P(E|not H) - `p_not_h = 1 - 0.20 = 0.80` is P(not H) - `p_e = 0.90 * 0.20 + 0.05 * 0.80 = 0.18 + 0.04 = 0.22` is the **evidence** P(E), computed via the law of total probability - The return value is `(0.90 * 0.20) / 0.22 = 0.18 / 0.22 = 0.8182` **Output:** `Result: 0.8182` Despite the low prior (20%), the strong likelihood ratio (0.90/0.05 = 18) causes a dramatic update to 81.82%. The evidence is 18 times more likely under the hypothesis than under the alternative, strongly supporting the hypothesis.Question 24
The following code has a subtle bug that leads to incorrect results in certain cases. Identify the bug and explain how to fix it.
def portfolio_ev(trades: list[dict]) -> float:
"""Calculate total portfolio expected value."""
total = 0.0
for trade in trades:
p = trade["market_price"]
q = trade["true_prob"]
size = trade["position_size"]
if q > p: # Only count positive EV trades
ev = (q - p) * size
total += ev
return total
trades = [
{"market_price": 0.40, "true_prob": 0.50, "position_size": 100},
{"market_price": 0.60, "true_prob": 0.55, "position_size": 100},
{"market_price": 0.30, "true_prob": 0.35, "position_size": 100},
]
print(f"Portfolio EV: ${portfolio_ev(trades):.2f}")
Answer
**The bug:** The function only counts trades with positive EV (the `if q > p` condition), ignoring trades with negative EV. But if the trader is **holding** all three positions, the negative EV trades still contribute to the portfolio's total expected value. In this example, Trade 2 has negative EV because the trader is buying at $0.60 but believes the true probability is only 0.55: EV = (0.55 - 0.60) * 100 = -$5.00. The current code ignores this loss. **Correct output with the bug:** The code reports only the positive EV trades: (0.50 - 0.40) * 100 + (0.35 - 0.30) * 100 = $10.00 + $5.00 = $15.00. **Correct output without the bug:** Including all trades: $10.00 + (-$5.00) + $5.00 = $10.00. **Fix:** Remove the `if q > p` condition. The function should sum EV for ALL positions held:def portfolio_ev(trades: list[dict]) -> float:
total = 0.0
for trade in trades:
ev = (trade["true_prob"] - trade["market_price"]) * trade["position_size"]
total += ev
return total
Question 25
What will the following simulation demonstrate as n_flips increases from 10 to 10,000? Explain the output pattern.
import numpy as np
def simulate(n_flips: int, true_p: float = 0.6, seed: int = 42) -> None:
rng = np.random.default_rng(seed)
flips = rng.random(n_flips) < true_p
running_avg = np.cumsum(flips) / np.arange(1, n_flips + 1)
print(f"n={n_flips:>6d}: "
f"final avg={running_avg[-1]:.4f}, "
f"max deviation={np.max(np.abs(running_avg - true_p)):.4f}")
for n in [10, 100, 1000, 10000]:
simulate(n)