Q: Which formula correctly expresses the general multiplication rule for any two events $$A$$ and $$B$$?

$$P(A \text{ and } B) = P(A)P(B|A)$$

Q: Given $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \text{ and } B) = 0.15$$, are $$A$$ and $$B$$ independent?

Yes, because $$P(A \text{ and } B) = P(A)P(B)$$

Q: A survey finds that $$11\% of $$the population smokes. The probability that a smoker will get lung cancer is $$0.34$$, and the probability that a non-smoker will get lung cancer is $$0.03. $$Are 'Smoker' and 'Lung cancer' independent?

No, because $$P(\text{Lung cancer}|\text{Smoker}) \neq P(\text{Lung cancer})$$

Q: Which of the following is the correct formula for conditional probability $$P(A|B)$$?

$$P(A|B) = \dfrac{P(A \text{ and } B)}{P(B)}$$

Q: A Dalmatian is hearing impaired with probability $$0.28$$ and blue-eyed with probability $$0.11. $$The probability that a Dalmatian is both hearing impaired and blue-eyed is $$0.05. $$Are these two traits independent?

No, because $$0.28 \times 0.11 \neq 0.05$$

Question 1

Which of the following best describes a marginal probability?

Accepted Answer

The probability of a single event without consideration of any other event.

Question 2

Suppose the probability that a randomly selected student is a senior is $$0.20$$, and the probability that a student is both a computer science major and a senior is $$0.03. $$What is the conditional probability that a student is a computer science major given that the student is a senior?

Accepted Answer

$$0.15$$

Question 3

If $$P(A) = 0.5$$, $$P(B) = 0.4$$, and $$A$$ and $$B$$ are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.2$$

Question 4

Which formula correctly expresses the general multiplication rule for any two events $$A$$ and $$B$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A)P(B|A)$$

Question 5

A blood donation center finds that the probability a random visitor is type O is $$0.45. If $$two unrelated visitors come in, what is the probability both are type O?

Accepted Answer

$$0.2025$$

Question 6

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

Accepted Answer

$$P(A|B) = P(A)$$

Question 7

In a diagnostic test for HIV-AIDS, the sensitivity is $$0.9985$$ and the specificity is $$0.9940. If $$the disease prevalence is $$0.01$$, what is the probability that a randomly selected adult who tests positive actually has HIV-AIDS (the positive predictive value)?

Accepted Answer

$$0.627$$

Question 8

Given $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.15$$, are $$A$$ and $$B$$ independent?

Accepted Answer

Yes, because $$P(A 	ext{ and } B) = P(A)P(B)$$

Question 9

A survey finds that $$11\% of $$the population smokes. The probability that a smoker will get lung cancer is $$0.34$$, and the probability that a non-smoker will get lung cancer is $$0.03. $$Are 'Smoker' and 'Lung cancer' independent?

Accepted Answer

No, because $$P(	ext{Lung cancer}|	ext{Smoker}) 
eq P(	ext{Lung cancer})$$

Question 10

Which of the following is the correct formula for conditional probability $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \dfrac{P(A 	ext{ and } B)}{P(B)}$$

Question 11

Suppose $$P(A) = 0.6$$, $$P(B) = 0.3$$, and $$P(A 	ext{ and } B) = 0.18. $$What is $$P(B|A)$$?

Accepted Answer

$$0.3$$

Question 12

A company has 60 males and 40 females. 15 males and 4 females are in favor of high CEO salaries. What is the probability that a randomly selected individual is in favor of high CEO salaries?

Accepted Answer

$$0.19$$

Question 13

Using the same company data, what is the probability that a randomly selected male is in favor of high CEO salaries?

Accepted Answer

$$0.15$$

Question 14

If $$P(A|B) = P(A)$$, what does this imply about events $$A$$ and $$B$$?

Accepted Answer

They are independent.

Question 15

A diagnostic test for a disease has a sensitivity of $$0.99$$ and a specificity of $$0.98. If $$the disease prevalence is $$0.02$$, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes' theorem.)

Accepted Answer

Approximately $$0.50$$

Question 16

Which of the following is NOT a property of independent events?

Accepted Answer

$$P(A|B) = P(B|A)$$

Question 17

If $$P(A) = 0.7$$, $$P(B) = 0.2$$, and $$A$$ and $$B$$ are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.14$$

Question 18

A Dalmatian is hearing impaired with probability $$0.28$$ and blue-eyed with probability $$0.11. $$The probability that a Dalmatian is both hearing impaired and blue-eyed is $$0.05. $$Are these two traits independent?

Accepted Answer

No, because $$0.28 	imes 0.11 
eq 0.05$$

Question 19

Which of the following best describes the positive predictive value (PPV) of a diagnostic test?

Accepted Answer

The probability that a person with a positive test result actually has the disease.

Independence and Conditional Probabilities: Core Concepts and Applications

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