Q: A diagnostic test for a disease has sensitivity $$0.95$$ and specificity $$0.90. If $$the prevalence of the disease is $$0.02$$, what is the probability that a person who tests positive actually has the disease? (Use Bayes' Rule)

$$\frac{0.95 \times 0.02}{0.95 \times 0.02 + 0.10 \times 0.98} \approx 0.162$$

Q: Suppose you draw two cards from a deck without replacement. What is the probability that both are aces?

$$\frac{4}{52} \times \frac{3}{51} \approx 0.0045$$

Question 1

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the number of trials increases, the proportion of outcomes for any event approaches a particular number, but may fluctuate wildly in the short run.

Question 2

Suppose you roll a fair six-sided die. What is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{2}{6} = 0.33$$

Question 3

If events $$A$$ and $$B$$ are mutually exclusive, which of the following is true?

Accepted Answer

$$P(A \cap B) = 0$$

Question 4

A hospital records that 12% of all patients are admitted for surgical treatment, 18% for obstetrics, and 2% for both. What is the probability that a randomly chosen patient is admitted for either surgical treatment or obstetrics?

Accepted Answer

$$0.12 + 0.18 - 0.02 = 0.28$$

Question 5

Which formula represents the probability of the union of two events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Question 6

If you flip a fair coin 1,000,000 times, what will the proportion of heads converge to?

Accepted Answer

$$0.5$$

Question 7

Given $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.15$$, what is $$P(A \mid B)$$?

Accepted Answer

$$\frac{P(A \cap B)}{P(B)} = \frac{0.15}{0.5} = 0.3$$

Question 8

Which of the following statements is true about independent events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cap B) = P(A) \cdot P(B)$$

Question 9

A diagnostic test for a disease has sensitivity $$0.95$$ and specificity $$0.90. If $$the prevalence of the disease is $$0.02$$, what is the probability that a person who tests positive actually has the disease? (Use Bayes' Rule)

Accepted Answer

$$\frac{0.95 	imes 0.02}{0.95 	imes 0.02 + 0.10 	imes 0.98} \approx 0.162$$

Question 10

Which of the following best describes the difference between sensitivity and specificity in a medical test?

Accepted Answer

Sensitivity is the probability that a test correctly identifies a diseased individual as positive, while specificity is the probability that a test correctly identifies a healthy individual as negative.

Question 11

Suppose you draw two cards from a deck without replacement. What is the probability that both are aces?

Accepted Answer

$$\frac{4}{52} 	imes \frac{3}{51} \approx 0.0045$$

Question 12

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 13

Which of the following is NOT a property of mutually exclusive events?

Accepted Answer

If $$A$$ and $$B$$ are mutually exclusive, $$P(A \cap B) = P(A) \cdot P(B)$$

Question 14

A contingency table shows that 1000 patients have a pulmonary embolism and a D-dimer positive result, out of 10,000 total patients. What is the probability that a patient has both conditions?

Accepted Answer

$$\frac{1000}{10000} = 0.1$$

Question 15

Which of the following is the correct formula for Bayes' Rule?

Accepted Answer

$$P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}$$

Question 16

If the probability of rolling a 2 on a fair six-sided die is $$\frac{1}{6}$$, what is the probability of NOT rolling a 2?

Accepted Answer

$$1 - \frac{1}{6} = \frac{5}{6}$$

Question 17

Which interpretation of probability is described by the long-run relative frequency of an event in repeated trials?

Accepted Answer

Frequentist interpretation

Question 18

Suppose the probability that a randomly chosen student likes either coffee or tea (or both) is $$0.63. If $$the probability of liking coffee is $$0.28$$ and tea is $$0.42$$, what is the probability that a student likes both?

Accepted Answer

$$0.28 + 0.42 - 0.63 = 0.07$$

Probability Concepts, Rules, and Bayes' Theorem: Structured Study Notes

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