Q: Which formula correctly calculates the probability of 'at least one' occurrence of event $$A in$$ $$n$$ independent trials?

$$P(\text{at least one } A) = 1 - [1 - P(A)]^n$$

Q: A medical test for cancer has a 1% prevalence rate, an 80% true positive rate, and a 10% false positive rate. What is the probability that a person who tests positive actually has cancer? Use Bayes’ theorem.

$$\frac{0.01 \times 0.80}{0.01 \times 0.80 + 0.99 \times 0.10}$$

Q: A test for a rare disease has a 5% false positive rate and a 90% true positive rate. The disease prevalence is 2%. What is the probability that a person who tests positive actually has the disease? (Use Bayes’ theorem.)

$$\frac{0.02 \times 0.90}{0.02 \times 0.90 + 0.98 \times 0.05}$$

Question 1

Which of the following best describes the complement of the event 'at least one' occurrence of a specific event in a series of trials?

Accepted Answer

Getting no occurrences of the event.

Question 2

A study finds that 6% of damaged iPads were damaged by bags/backpacks. If 20 damaged iPads are randomly selected, what is the probability that at least one was damaged in a bag/backpack? Use the complement rule.

Accepted Answer

$$1 - 0.94$$^{20}$$

Question 3

Which formula correctly calculates the probability of 'at least one' occurrence of event $$A in$$ $$n$$ independent trials?

Accepted Answer

$$P(	ext{at least one } A) = 1 - [1 - P(A)]^n$$

Question 4

Given events $$A$$ and $$B$$, which formula gives the conditional probability $$P(B|A)$$?

Accepted Answer

$$P(B|A) = \frac{P(A \cap B)}{P(A)}$$

Question 5

In a pre-employment drug screening, 50 subjects use drugs and 45 of them test positive. What is $$P(	ext{positive test} | 	ext{subject uses drugs})$$?

Accepted Answer

$$\frac{45}{50}$$

Question 6

A medical test for cancer has a 1% prevalence rate, an 80% true positive rate, and a 10% false positive rate. What is the probability that a person who tests positive actually has cancer? Use Bayes’ theorem.

Accepted Answer

$$\frac{0.01 	imes 0.80}{0.01 	imes 0.80 + 0.99 	imes 0.10}$$

Question 7

Which of the following statements about conditional probability is TRUE?

Accepted Answer

Conditional probability requires knowledge that another event has already occurred.

Question 8

In the context of Bayes’ theorem, what is a 'prior probability'?

Accepted Answer

A probability calculated before any additional information is obtained.

Question 9

Suppose the probability that a randomly selected person has a disease is $$0.01. $$After a positive test result, the probability that the person has the disease increases to $$0.0748. $$What is $$0.0748$$ called?

Accepted Answer

Posterior probability

Question 10

A common error in interpreting conditional probabilities is called 'confusion of the inverse.' Which of the following illustrates this error?

Accepted Answer

Assuming $$P(A|B) = P(B|A)$$ for all events.

Question 11

If the probability that a test is positive given the subject uses drugs is $$0.9$$, and the probability that a subject uses drugs given a positive test is $$0.643$$, what does this difference illustrate?

Accepted Answer

Conditional probabilities depend on the direction of conditioning.

Question 12

A company wants to know the probability that at least one of its 10 randomly selected products is defective, given that the probability a product is not defective is $$0.98. $$What is the correct calculation?

Accepted Answer

$$1 - 0.98$$^{10}$$

Question 13

Which of the following is NOT a step in applying Bayes’ theorem to revise a probability?

Accepted Answer

Assume the prior and posterior probabilities are always equal.

Question 14

A test for a rare disease has a 5% false positive rate and a 90% true positive rate. The disease prevalence is 2%. What is the probability that a person who tests positive actually has the disease? (Use Bayes’ theorem.)

Accepted Answer

$$\frac{0.02 	imes 0.90}{0.02 	imes 0.90 + 0.98 	imes 0.05}$$

Question 15

Which scenario best illustrates the use of the complement rule in probability?

Accepted Answer

Calculating the probability of getting at least one heads in 5 coin tosses.

Question 16

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.15$$

Question 17

A hospital screens 1000 patients for a disease with a prevalence of 1%. The test has an 80% true positive rate and a 10% false positive rate. How many patients without the disease are expected to test positive?

Accepted Answer

99

Question 18

Which of the following is a real-world example of conditional probability?

Accepted Answer

The probability that a person has the flu given that they have a fever.

Chapter 4: Probability – Complements, Conditional Probability, and Bayes’ Theorem

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