📖 概率周测（10 题）

Quantitative Methods — Probability Module · Weekly Assessment

1. Test Instructions

2. Quick Reference: Key Formulas

3. 10 Test Questions

Question 1 (L115 · Probability Basics)

An analyst studies 100 stocks in a sector: 60 pay dividends (A), 40 beat earnings expectations (B), and 24 both pay dividends and beat earnings expectations. If a stock is selected at random, the probability that it pays dividends OR beats earnings expectations is:

A. 0.24 B. 0.76 C. 1.00 D. 0.52

Question 2 (L116 · Expected Value & Variance)

An investment strategy has the following return distribution:

The strategy's expected return and variance (in thousands² of yen) are closest to:

A. E=¥1,000, Var=4.6 B. E=¥1,000, Var=7.8 C. E=¥1,400, Var=6.6 D. E=¥1,400, Var=9.84

Question 3 (L117 · Conditional Probability & Bayes)

A fund company's research team consists of 30% senior analysts and 70% junior analysts. Historical data shows: senior analysts have an 80% forecast accuracy rate; junior analysts have a 60% forecast accuracy rate. A forecast is verified to be correct. The posterior probability that this forecast came from a senior analyst is closest to:

A. 0.24 B. 0.30 C. 0.36 D. 0.44

Question 4 (L117 · Bayes Application)

Among listed companies, the prior probability of earnings manipulation is 5%. An audit procedure has a 95% chance of issuing a "qualified opinion" when the company is manipulating, and a 10% chance of a false positive when the company is clean. If a company receives a qualified opinion, the probability that it is actually manipulating earnings is closest to:

A. 5.0% B. 33.3% C. 47.5% D. 95.0%

Question 5 (L118 · Permutations & Combinations)

A portfolio manager needs to select 5 stocks from a pool of 12 to construct an equal-weighted portfolio, and select 3 bonds from a pool of 8 to construct an equal-weighted portfolio. The total number of ways to construct both portfolios is:

A. C(12,5) + C(8,3) B. C(12,5) × C(8,3) C. P(12,5) × P(8,3) D. C(12,5) × P(8,3)

Question 6 (L120 · Normal Distribution)

The annual returns of stocks in a certain sector follow N(10%, 15%²). Using the 68-95-99.7 rule, the probability that annual returns fall between −5% and 25% is approximately:

A. 50% B. 68% C. 95% D. 99.7%

Question 7 (L121 · Z-Score & Normal Probabilities)

Continuing from Q6, the probability that the annual return is negative is closest to:

A. 16% B. 25% C. 33% D. 50%

Question 8 (L116+L120 · Covariance & Correlation)

A portfolio contains assets P and Q, with σₚ = 18%, σQ = 24%, and ρₚQ = −0.2. The covariance between P and Q is:

A. −0.00864 B. −0.00432 C. 0.00432 D. 0.00864

Question 9 (L122 · Lognormal Distribution)

A stock price S follows a lognormal distribution, with ln S ~ N(3.5, 0.3²). The median and mean of the stock price are closest to:

A. Median = e³·⁵ ≈ 33.12; Mean = e³·⁵⁴⁵ ≈ 34.64 B. Median = e³·⁵ ≈ 33.12; Mean = e³·⁵ = 33.12 C. Median = e³·² ≈ 24.53; Mean = e³·⁵ ≈ 33.12 D. Cannot be determined; insufficient data

Question 10 (L118+L115 · Combined Application)

An investment committee consists of 3 economists and 5 analysts. Four members are randomly selected to form a task force. The expression for the probability that exactly 2 economists are selected is:

A. C(3,2) × C(5,2) / C(8,4) B. C(3,2) / C(8,4) C. C(5,2) / C(8,4) D. P(3,2) × P(5,2) / P(8,4)

4. Answers & Explanations

[Q1 Answer] B — 0.76

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.60 + 0.40 - 0.24 = 0.76$$

💡 Addition rule: subtract the intersection to avoid double-counting. Verification: stocks with neither attribute = 100 − (60+40−24) = 24 → P = 0.24. Correct.

[Q2 Answer] D — E=¥1,400, Var=9.84

Working in thousands of yen (values: 5, 2, −3):

$$\begin{aligned} E(X) &= 0.30 \times 5 + 0.40 \times 2 + 0.30 \times (-3) = 1.5 + 0.8 - 0.9 = 1.4 \text{ (thousand yen)} \[8pt] E(X^2) &= 0.30 \times 25 + 0.40 \times 4 + 0.30 \times 9 = 7.5 + 1.6 + 2.7 = 11.8 \[8pt] \text{Var}(X) &= E(X^2) - [E(X)]^2 = 11.8 - 1.96 = 9.84 \text{ (thousands²)} \end{aligned}$$

💡 Using Var = E(X²) − [E(X)]² is the fastest computational approach.

[Q3 Answer] C — 0.36 (exact: 36.4%)

$$\begin{aligned} P(\text{Correct}) &= 0.30 \times 0.80 + 0.70 \times 0.60 = 0.24 + 0.42 = 0.66 \[4pt] P(\text{Senior} \mid \text{Correct}) &= \frac{P(\text{Correct} \mid \text{Senior}) \cdot P(\text{Senior})}{P(\text{Correct})} = \frac{0.80 \times 0.30}{0.66} \[4pt] &= \frac{0.24}{0.66} \approx 0.3636 \approx 0.36 \end{aligned}$$

💡 Posterior 36.4% > Prior 30%: the evidence "forecast was correct" increases confidence in a senior analyst, but modestly.

[Q4 Answer] B — 33.3%

$$\begin{aligned} P(\text{Qualified}) &= 0.95 \times 0.05 + 0.10 \times 0.95 = 0.0475 + 0.095 = 0.1425 \[4pt] P(\text{Fraud} \mid \text{Qualified}) &= \frac{0.95 \times 0.05}{0.1425} = \frac{0.0475}{0.1425} \approx 0.3333 \end{aligned}$$

💡 Classic Base Rate Neglect example: despite a seemingly accurate audit test, the low prior probability of fraud (5%) means that even after receiving a qualified opinion, the probability of actual fraud is only ~33%. High-frequency CFA question on cognitive biases.

[Q5 Answer] B — C(12,5) × C(8,3)

Equal-weighted portfolios → order does not matter → use combinations. Stocks and bonds are two independent selection steps → multiplication principle.

$$\text{Total ways} = C(12,5) \times C(8,3) = 792 \times 56 = 44,352$$

💡 Two independent steps → multiply. Portfolio construction (equal-weighted) → ignore order → combinations.

[Q6 Answer] B — 68%

μ = 10%, σ = 15%: - −5% = 10% − 15% = μ − σ - 25% = 10% + 15% = μ + σ

68-95-99.7 rule: μ ± σ covers approximately 68% of the data.

💡 First convert bounds to standard deviations from the mean, then apply the rule.

[Q7 Answer] B — 25%

Negative return means X < 0%.

0% = 10% − 0.67×15% ≈ μ − 0.67σ

z = (0 − 10)/15 = −0.67, P(Z < −0.67) ≈ 25%.

💡 Key z-score tail probabilities to memorize: P(Z < −0.67) ≈ 25%, P(Z < −1) ≈ 16%, P(Z < −1.65) ≈ 5%, P(Z < −1.96) ≈ 2.5%.

[Q8 Answer] A — −0.00864

$$\text{Cov}(P,Q) = \rho_{PQ} \cdot \sigma_P \cdot \sigma_Q = (-0.2) \times 0.18 \times 0.24 = -0.00864$$

💡 Negative correlation → negative covariance → better diversification benefit. Covariance = Correlation coefficient × product of standard deviations.

[Q9 Answer] A — Median = e³·⁵ ≈ 33.12; Mean = e³·⁵⁴⁵ ≈ 34.64

$$\begin{aligned} \text{Median}(S) &= e^{\mu} = e^{3.5} \approx 33.12 \[4pt] E(S) &= e^{\mu + \sigma^2/2} = e^{3.5 + 0.09/2} = e^{3.5 + 0.045} = e^{3.545} \approx 34.64 \end{aligned}$$

💡 Mean > Median (34.64 > 33.12) reflects the right-skewed nature of the lognormal distribution. Key distinction: median uses only μ; mean adds the σ²/2 adjustment term.

[Q10 Answer] A — C(3,2) × C(5,2) / C(8,4)

Hypergeometric distribution: population of 8 (3 economists + 5 analysts), draw 4, exactly 2 economists.

$$\begin{aligned} \text{Numerator} &= C(3,2) \times C(5,2) = 3 \times 10 = 30 \[4pt] \text{Denominator} &= C(8,4) = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \[4pt] \text{Probability} &= \frac{30}{70} = \frac{3}{7} \approx 0.4286 \end{aligned}$$

💡 Sampling without replacement from a finite population → hypergeometric distribution. "Exactly k of type X" = C(pick k from type X) × C(pick n−k from others) / C(pick n from total).

5. Score Evaluation & Study Recommendations

Scoring Rubric

Topic–Question Mapping

Top 3 High-Frequency Errors

1. Bayes' Theorem (Q3, Q4): ~30% of candidates forget to compute the total probability P(A) as the denominator first
2. Permutations vs Combinations (Q5, Q10): the deciding criterion is always "does order matter?"
3. Lognormal Mean (Q9): confusing e^μ (median) with e^(μ+σ²/2) (mean)

📚 Probability module complete. Next module: L125 begins Sampling and Estimation.