Quiz: Conditional Probability and Bayes' Theorem

Test your understanding of conditional probability, Bayes' theorem, and their applications. Try to answer each question before revealing the answer.

1. What does the notation $P(A \mid B)$ mean?

(a) The probability of A and B occurring together (b) The probability of A occurring, given that B is known to have occurred (c) The probability of A or B occurring (d) The probability of B occurring after A

Answer **(b) The probability of A occurring, given that B is known to have occurred.** The vertical bar "|" means "given that." $P(A \mid B)$ restricts the sample space to only those outcomes where B has occurred, then asks what fraction of those also satisfy A.

2. In a contingency table of 400 patients, 80 are smokers and 120 have a respiratory condition. Of the 80 smokers, 48 have the respiratory condition. What is $P(\text{respiratory} \mid \text{smoker})$?

(a) 48/400 = 0.12 (b) 48/80 = 0.60 (c) 48/120 = 0.40 (d) 80/400 = 0.20

Answer **(b) 48/80 = 0.60** When conditioning on "smoker," the denominator becomes the number of smokers (80), not the total (400). Among the 80 smokers, 48 have a respiratory condition: 48/80 = 0.60.

3. Using the same data from Question 2, what is $P(\text{smoker} \mid \text{respiratory})$?

(a) 48/400 = 0.12 (b) 48/80 = 0.60 (c) 48/120 = 0.40 (d) 80/120 = 0.67

Answer **(c) 48/120 = 0.40** Now we condition on "respiratory condition," so the denominator is 120 (all people with the condition). Of those, 48 are smokers: 48/120 = 0.40. Note that $P(\text{respiratory} \mid \text{smoker}) = 0.60 \neq P(\text{smoker} \mid \text{respiratory}) = 0.40$ — they are different probabilities answering different questions.

4. Which of the following best describes the prosecutor's fallacy?

(a) Using probability to determine guilt or innocence (b) Confusing $P(\text{evidence} \mid \text{innocent})$ with $P(\text{innocent} \mid \text{evidence})$ (c) Ignoring evidence that supports the defense (d) Using unreliable statistical methods in court

Answer **(b) Confusing $P(\text{evidence} \mid \text{innocent})$ with $P(\text{innocent} \mid \text{evidence})$.** The prosecutor's fallacy is the specific error of treating $P(A \mid B)$ as if it were $P(B \mid A)$. A prosecutor might say "the probability of this evidence occurring by chance is 1 in a million, so the probability of innocence is 1 in a million" — but these are completely different probabilities.

5. A disease has a prevalence of 2%. A diagnostic test has 95% sensitivity and 90% specificity. What is the false positive rate?

(a) 2% (b) 5% (c) 10% (d) 95%

Answer **(c) 10%** The false positive rate = 1 - specificity = 1 - 0.90 = 0.10 = 10%. This means 10% of healthy people will incorrectly test positive.

6. Using the same disease from Question 5 (prevalence 2%, sensitivity 95%, specificity 90%), if a randomly selected person tests positive, approximately what is the probability they actually have the disease?

(a) About 16% (b) About 50% (c) About 90% (d) About 95%

Answer **(a) About 16%** $P(\text{disease} \mid \text{positive}) = \frac{0.95 \times 0.02}{0.95 \times 0.02 + 0.10 \times 0.98} = \frac{0.019}{0.019 + 0.098} = \frac{0.019}{0.117} \approx 0.162$ Despite the test having 95% sensitivity, the PPV is only about 16% because the disease is rare (2%) and false positives from the large healthy population (10% of 98%) overwhelm the true positives.

7. In the natural frequency approach to Bayes' theorem, why do we start by imagining a large group of people (e.g., 10,000)?

(a) Because Bayes' theorem only works with large samples (b) Because it's easier for our brains to reason about frequencies than decimal probabilities (c) Because small numbers produce rounding errors (d) Because the law of large numbers requires large samples

Answer **(b) Because it's easier for our brains to reason about frequencies than decimal probabilities.** Research by Gerd Gigerenzer and others shows that people — including physicians — reason about probabilities much more accurately when information is presented as natural frequencies ("2 out of 1,000") rather than probabilities ("0.2%"). Our brains evolved to track counts, not decimals.

8. Which of the following is Bayes' theorem?

(a) $P(A \mid B) = P(A) + P(B) - P(A \text{ and } B)$ (b) $P(A \mid B) = P(A) \times P(B)$ (c) $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$ (d) $P(A \mid B) = 1 - P(B \mid A)$

Answer **(c) $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$** Bayes' theorem provides the mathematical bridge between P(B|A) and P(A|B). The other options are: (a) the addition rule, (b) the independence multiplication rule, and (d) not a valid probability identity.

9. In Bayes' theorem, the "prior probability" refers to:

(a) The probability calculated after seeing evidence (b) The probability of the evidence regardless of the hypothesis (c) The probability of the hypothesis before seeing any evidence (d) The probability of getting a false positive

Answer **(c) The probability of the hypothesis before seeing any evidence.** The prior is what you believed before the new evidence arrived. In medical testing, it's the prevalence (base rate) of the disease. The prior gets updated to the posterior through Bayes' theorem.

10. A screening test for a rare condition (prevalence 0.01%) has 99% sensitivity and 99% specificity. Out of 1,000,000 people tested, approximately how many will test positive?

(a) About 99 (b) About 1,000 (c) About 10,000 (d) About 10,099

Answer **(d) About 10,099** Out of 1,000,000 people: 100 have the condition (0.01%), and 999,900 don't. - True positives: 100 × 0.99 = 99 - False positives: 999,900 × 0.01 = 9,999 - Total positives: 99 + 9,999 = 10,098 ≈ 10,099 False positives outnumber true positives about 100 to 1 because the condition is so rare.

11. Sensitivity is best described as:

(a) The probability a healthy person tests negative (b) The probability a person with the disease tests positive (c) The probability a positive test result is correct (d) The probability a negative test result is correct

Answer **(b) The probability a person with the disease tests positive.** Sensitivity = $P(\text{positive} \mid \text{disease})$, also called the true positive rate. It measures how well the test catches people who actually have the condition. Note that sensitivity is a property of the test, not a measure of what a positive result means (that's the PPV).

12. Two events A and B are independent if and only if:

(a) $P(A \text{ and } B) = 0$ (b) $P(A \mid B) = P(A)$ (c) $P(A) + P(B) = 1$ (d) $P(A \mid B) = P(B \mid A)$

Answer **(b) $P(A \mid B) = P(A)$** Independence means knowing B occurred doesn't change the probability of A. If $P(A \mid B) = P(A)$, then conditioning on B has no effect — the events are independent. Option (a) describes mutually exclusive events, not independent events.

13. A spam filter calculates $P(\text{spam} \mid \text{words in email})$. If an email contains the word "free" and the filter knows that $P(\text{"free"} \mid \text{spam}) = 0.60$ and $P(\text{"free"} \mid \text{legitimate}) = 0.04$, what is the likelihood ratio for the word "free"?

(a) 0.04/0.60 = 0.067 (b) 0.60/0.04 = 15 (c) 0.60 × 0.04 = 0.024 (d) 0.60 + 0.04 = 0.64

Answer **(b) 0.60/0.04 = 15** The likelihood ratio = $P(\text{evidence} \mid \text{hypothesis}) / P(\text{evidence} \mid \text{not hypothesis})$. A ratio of 15 means the word "free" is 15 times more likely to appear in spam than in legitimate email — making it strong evidence of spam.

14. What is the "base rate fallacy"?

(a) Calculating the base rate incorrectly (b) Using an incorrect base rate in Bayes' theorem (c) Ignoring the prior probability (base rate) when evaluating evidence (d) Confusing the base rate with the false positive rate

Answer **(c) Ignoring the prior probability (base rate) when evaluating evidence.** The base rate fallacy occurs when people focus on the specific evidence (e.g., "the test is 99% accurate") and ignore the general prevalence (e.g., "the disease affects 1 in 10,000 people"). The base rate is often the most important factor in determining what evidence actually means.

15. In a tree diagram for a disease screening problem, which path represents a false positive?

(a) Has disease → Tests positive (b) Has disease → Tests negative (c) No disease → Tests positive (d) No disease → Tests negative

Answer **(c) No disease → Tests positive** A false positive occurs when the test says "positive" but the person doesn't actually have the disease. On the tree diagram, this is the branch where you go right at the first split (no disease) and then left at the second split (test positive).

16. The law of total probability states that $P(B) = P(B \mid A) \cdot P(A) + P(B \mid A') \cdot P(A')$. In Bayes' theorem, this expression is used as:

(a) The numerator (b) The denominator (c) The prior (d) The likelihood

Answer **(b) The denominator** In Bayes' theorem, $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$, and $P(B)$ in the denominator is calculated using the law of total probability. It represents the overall probability of the evidence, combining all possible pathways through which B can occur.

17. If a positive test result gives a posterior probability of 5% for a disease, and a second independent test is also positive, what happens to the probability?

(a) It stays at 5% because the tests are independent (b) It increases — the 5% becomes the new prior, and the second positive test provides additional evidence (c) It becomes 10% (5% × 2) (d) It becomes 0.25% (5% × 5%)

Answer **(b) It increases — the 5% becomes the new prior, and the second positive test provides additional evidence.** This is the power of sequential Bayesian updating. The posterior from the first test (5%) becomes the prior for the second test. A second positive result, processed through Bayes' theorem again, will increase the probability further. The exact value depends on the test's sensitivity and specificity.

18. Which statement about the positive predictive value (PPV) is TRUE?

(a) PPV depends only on the test's accuracy (sensitivity and specificity) (b) PPV is always higher than sensitivity (c) PPV depends on both the test's accuracy AND the prevalence of the condition (d) PPV and sensitivity are the same thing

Answer **(c) PPV depends on both the test's accuracy AND the prevalence of the condition.** This is a critical insight: sensitivity and specificity are properties of the *test* alone, but PPV (the probability of actually having the condition given a positive test) depends on both the test AND the base rate. A highly accurate test can have a very low PPV when the condition is rare.

19. A weather app says there's a 40% chance of rain. You look outside and see dark clouds. Using Bayesian reasoning, which of the following best describes how you should update your probability?

(a) It should stay at 40% because the app already considered weather patterns (b) It should go up, because dark clouds are more likely when rain is coming than when it isn't (c) It should go to 100% because you can see evidence of rain (d) It should go down because you haven't seen actual rain yet

Answer **(b) It should go up, because dark clouds are more likely when rain is coming than when it isn't.** In Bayesian terms: $P(\text{dark clouds} \mid \text{rain})$ is high, and $P(\text{dark clouds} \mid \text{no rain})$ is lower. This likelihood ratio greater than 1 means the evidence of dark clouds should increase your probability estimate above the 40% prior. (Assuming the app's forecast didn't already factor in your current sky observation.)

20. Which of the following is NOT a real-world application of Bayes' theorem?

(a) Spam filters classifying emails as spam or legitimate (b) Calculating the arithmetic mean of a dataset (c) Medical tests determining the probability of disease given a positive result (d) AI language models predicting the next word in a sentence

Answer **(b) Calculating the arithmetic mean of a dataset** The arithmetic mean is a summary statistic — it doesn't involve conditional probability or Bayesian updating. All other options involve conditional probability at their core: spam filters calculate P(spam | words), medical testing involves P(disease | positive), and language models calculate P(next word | previous words).

Scoring Guide

Score Interpretation Recommendation
18-20 Excellent — you've mastered Bayes' theorem and conditional probability Move confidently to Chapter 10
15-17 Strong — solid understanding with minor gaps Review the sections for questions you missed, then proceed
12-14 Developing — core concepts are there but details need work Re-read Sections 9.5-9.7 (tree diagrams and natural frequencies) and redo exercises C.1-C.4
9-11 Needs review — revisit the chapter before proceeding Focus on Sections 9.2-9.3 (conditional probability basics) and 9.6-9.7 (Bayes' theorem and natural frequencies)
Below 9 Significant gaps — this material is critical for later chapters Re-read the full chapter, work through all Part B and C exercises, then retake the quiz