Q: A medical test for a disease has a 95% chance of correctly identifying a diseased person and a 5% chance of incorrectly identifying a healthy person as diseased. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease?

$$\frac{0.95 \times 0.01}{0.95 \times 0.01 + 0.05 \times 0.99}$$

Question 1

Which of the following best describes a sample space in probability theory?

Accepted Answer

A set containing all possible outcomes of an experiment.

Question 2

If $$A$$ and $$B$$ are mutually exclusive events, which equation correctly represents $$P(A \cup B)$$?

Accepted Answer

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

Question 3

A box contains 3 red balls, 2 green balls, and 1 blue ball. If one ball is drawn at random, what is the probability that it is not red?

Accepted Answer

$$\frac{1}{2}$$

Question 4

Which of the following is NOT an axiom of probability?

Accepted Answer

$$P(E) = P(E^c)$$

Question 5

Given events $$A$$ and $$B in a $$sample space, which formula gives the probability that either $$A or$$ $$B$$ occurs?

Accepted Answer

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

Question 6

A factory produces bulbs, 6 of which are white and 3 are black. If two bulbs are drawn at random without replacement, what is the probability that one is white and one is black?

Accepted Answer

$$\frac{6}{9} \cdot \frac{3}{8} + \frac{3}{9} \cdot \frac{6}{8}$$

Question 7

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.7$$

Question 8

Which of the following statements about independent events is correct?

Accepted Answer

If $$A$$ and $$B$$ are independent, then $$P(A \cap B) = P(A) \cdot P(B)$$

Question 9

A medical test for a disease has a 95% chance of correctly identifying a diseased person and a 5% chance of incorrectly identifying a healthy person as diseased. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease?

Accepted Answer

$$\frac{0.95 	imes 0.01}{0.95 	imes 0.01 + 0.05 	imes 0.99}$$

Question 10

If $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$, what does $$P(A|B)$$ represent?

Accepted Answer

The probability that event $$A$$ occurs given that event $$B$$ has occurred.

Question 11

A committee of size 3 is to be selected from a group of 5 men and 7 women. What is the probability that the committee consists of 2 men and 1 woman?

Accepted Answer

$$\frac{\binom{5}{2} \binom{7}{1}}{\binom{12}{3}}$$

Question 12

Which of the following is the correct formula for the odds in favor of event $$A$$?

Accepted Answer

$$\frac{P(A)}{1 - P(A)}$$

Question 13

Suppose $$A$$ and $$B$$ are independent events with $$P(A) = 0.3$$ and $$P(B) = 0.5. $$What is $$P(A \cap B)$$?

Accepted Answer

$$0.15$$

Question 14

If $$P(A) = 0.6$$ and $$P(B) = 0.4$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cap B)$$?

Accepted Answer

$$0$$

Question 15

A student answers a multiple-choice question with 4 options, only one of which is correct. If the student knows the answer with probability $$0.7$$ and guesses otherwise, what is the probability that the student answers correctly?

Accepted Answer

$$0.7 + 0.3 	imes 0.25$$

Question 16

Which of the following is the correct statement of Bayes' formula for events $$A$$ and $$B$$?

Accepted Answer

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

Question 17

If $$E$$ and $$F$$ are independent events, which of the following is true?

Accepted Answer

$$P(E \cap F) = P(E) \cdot P(F)$$

Question 18

A system consists of 4 components, each of which functions independently with probability $$p. $$What is the probability that at least one component functions?

Accepted Answer

$$1 - (1-p)^4$

Question 19

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12$$, what is $$P(A|B)$$?

Accepted Answer

$$0.12 / 0.4 = 0.3$$

Elements of Probability: Structured Study Notes

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