Skip to main content

Module Quiz

Complete this quiz after finishing all concept and practice pages.

Current Module Questions​

Question 1: Sample Space Discipline​

Two fair coin flips are made. Why is {HH, HT, TH, TT} a better sample space than {0 heads, 1 head, 2 heads} for equally likely reasoning?

Answer: The ordered outcomes {HH, HT, TH, TT} are the atomic outcomes of the experiment and are equally likely. The summary values {0,1,2} are not equally likely because 1 head corresponds to two different atomic outcomes.

Question 2: Complement Thinking​

Five independent components each fail with probability 0.01. What is the probability that at least one fails?

Answer: 1 - 0.99^5.

Solution Walkthrough:

  1. "At least one fails" is the complement of "none fail."
  2. Since failures are independent, P(none fail) = 0.99^5.
  3. Subtract from 1.

Question 3: Conditional Probability​

Two cards are drawn without replacement. Given the first card is an ace, what is the probability the second is also an ace?

Answer: 3/51 = 1/17.

Question 4: Bayes and Base Rates​

A rare condition affects 2% of users. A detector flags a true case 95% of the time and falsely flags a healthy case 10% of the time. Write the posterior probability of the condition given a flag.

Answer:
P(C | F) = 0.95(0.02) / (0.95(0.02) + 0.10(0.98)).

Question 5: Independence vs Disjointness​

Can two positive-probability disjoint events be independent?

Answer: No. If A and B are disjoint, then P(A intersect B)=0, but for positive-probability independent events we would need P(A intersect B)=P(A)P(B)>0.

Question 6: Random Variables​

What is the difference between X and the event X = 3?

Answer: X is the random variable, a function from outcomes to numbers. X = 3 is an event: the set of outcomes for which the function takes the value 3.

Question 7: Distribution Selection​

Which model is more appropriate for the number of defective items in a sample of 8 drawn without replacement from a lot with a fixed number of defects: binomial or hypergeometric?

Answer: Hypergeometric, because sampling is without replacement from a finite population.

Question 8: Expectation​

A random variable takes value 0 with probability 0.7 and value 5 with probability 0.3. What is its expectation?

Answer: E[X] = 0(0.7) + 5(0.3) = 1.5.

Question 9: Indicator Variables​

Why is E[I_A] = P(A) for an indicator variable?

Answer: Because I_A equals 1 on event A and 0 otherwise, so its probability-weighted average is exactly the probability that A occurs.

Question 10: Variance​

What kind of information does variance provide that expectation alone does not?

Answer: Variance measures spread around the mean. It tells how unstable or volatile the values are, which expectation alone cannot capture.

Question 11: Continuous Variables​

Why is P(X = c) = 0 for a continuous random variable?

Answer: Because continuous probabilities come from area under a density, and a single point has zero width and therefore zero area.

Question 12: Standardization​

What does the standardized value z = (x - mu)/sigma mean?

Answer: It tells how many standard deviations the value x is above or below the mean.

Question 13: Law of Large Numbers​

What does the law of large numbers say about sample averages?

Answer: Over many i.i.d. trials, the sample average approaches the true mean.

Question 14: Central Limit Theorem​

What does the central limit theorem add beyond the law of large numbers?

Answer: It describes the approximate distribution of the normalized sample mean for large n, showing that it becomes approximately Normal under broad conditions.

Question 15: Confidence Language​

Why is the statement "there is a 95% probability the fixed parameter lies in this computed 95% confidence interval" usually considered sloppy in the frequentist setting?

Answer: Because after the interval is computed, the parameter is fixed and the interval is fixed. The 95% refers to the long-run success rate of the interval-producing procedure, not a probability on the already fixed parameter.

Interleaved Review Questions​

Prior Module Question 1​

Why is a proof by examples not enough for a universal claim?

Answer: Because checking some instances does not rule out counterexamples elsewhere.

Prior Module Question 2​

What is the cleanest structure for proving two sets are equal?

Answer: Show double inclusion: A subseteq B and B subseteq A.

Prior Module Question 3​

What is the difference between ordered and unordered choice?

Answer: Ordered choice treats different arrangements as different outcomes; unordered choice does not.

Prior Module Question 4​

When is stars and bars relevant?

Answer: When counting nonnegative integer solutions or distributions of identical units across categories.

Prior Module Question 5​

What does induction prove once the base case and inductive step are complete?

Answer: That the statement holds for all values in the intended domain generated by the induction structure.

Self-Assessment and Remediation​

Mastery Level (90-100% correct):

  • Ready to advance with confidence.

Proficient Level (75-89% correct):

  • Review only the missed concept pages and redo two problems of each missed type.

Developing Level (60-74% correct):

  • Rework the practice pages, especially Bayes/conditioning and expectation/variance.

Insufficient Level (<60% correct):

  • Return to the concept sequence and rebuild the model-first habits before advancing.