Q: Which of the following best describes the multiplication rule for independent events?

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

Question 1

Suppose a medical test for a rare disease has a 99% specificity (correctly identifies healthy people) and an 80% sensitivity (correctly identifies sick people). If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease? (Use Bayes' Theorem.)

Accepted Answer

$$\frac{8}{107}$$

Question 2

If $$P(A) = 0.3$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.15$$

Question 3

Which of the following best describes the multiplication rule for independent events?

Accepted Answer

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

Question 4

If the probability that it rains on any given day is $$0.2$$, what is the probability that it does not rain?

Accepted Answer

$$0.8$$

Question 5

A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

Accepted Answer

$$26^3$$ 	imes $$10^2$$

Question 6

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

Accepted Answer

$$0.2$$

Question 7

Which formula correctly expresses the probability of at least one success in $$n$$ independent trials, each with probability $$p of $$failure?

Accepted Answer

$$1 - p^n$$

Question 8

If $$P(A|B) = 0.7$$ and $$P(B) = 0.4$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.28$$

Question 9

A bag contains 5 red balls and 7 blue balls. If two balls are drawn one after another with replacement, what is the probability both are red?

Accepted Answer

$$\left(\frac{5}{12}\$$right)^2$$

Question 10

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

Accepted Answer

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

Question 11

If $$P(C) = 0.01$$, $$P(T|C) = 0.8$$, $$P(T|C^c) = 0.01$$, and $$P(C^c) = 0.99$$, what is $$P(T)$$?

Accepted Answer

$$0.0179$$

Question 12

A student guesses on a multiple-choice question with 4 options. What is the probability they get it wrong?

Accepted Answer

$$\frac{3}{4}$$

Question 13

If $$P(A) = 0.6$$ and $$P(B) = 0.3$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.9$$

Question 14

Which of the following is NOT a requirement for using the multiplication rule for independent events?

Accepted Answer

Events must be mutually exclusive.

Question 15

If you flip a fair coin 3 times, what is the probability of getting at least one head?

Accepted Answer

$$1 - (0.5)^3$

Probability Rules and Counting Principles in Introductory Statistics

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