Q: Suppose you toss two fair coins. What is the probability of observing exactly one head?

$$P(\text{exactly one head}) = \frac{2}{4} = 0.5$$

Q: A hospital reports that 12% of patients are admitted for surgery, 16% for obstetrics, and 2% for both. What is the probability that a randomly admitted patient is admitted for either surgery or obstetrics or both?

$$P(\text{surgery} \cup \text{obstetrics}) = 0.12 + 0.16 - 0.02 = 0.26$$

Q: Given the following probabilities from a manufacturing process: $$P(DD) = 0.01$$, $$P(DN) = 0.045$$, $$P(ND) = 0.045$$, $$P(NN) = 0.9. $$What is the probability of observing at least one defective smartphone in a sample of two?

$$P(\text{at least one defective}) = 0.01 + 0.045 + 0.045 = 0.1$$

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

$$P(A \cap B) = 0.5 \times 0.4 = 0.2$$

Q: A study finds that 55% of executives have cheated at golf, and 20% have both cheated at golf and lied in business. What is the probability that an executive who has cheated at golf has also lied in business?

$$P(\text{lied in business} | \text{cheated at golf}) = \frac{0.20}{0.55} = 0.364$$

Question 1

Which of the following best describes the difference between a simple event and a compound event in probability theory?

Accepted Answer

A simple event contains only one sample point, while a compound event contains two or more sample points.

Question 2

Suppose you toss two fair coins. What is the probability of observing exactly one head?

Accepted Answer

$$P(	ext{exactly one head}) = \frac{2}{4} = 0.5$$

Question 3

Given the following probabilities for two events: $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.1. $$What is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = 0.3 + 0.4 - 0.1 = 0.6$$

Question 4

If the probability of an event $$A is$$ $$0.7$$, what is the probability of its complement $$A^C$$?

Accepted Answer

$$P(A^C) = 1 - 0.7 = 0.3$$

Question 5

A hospital reports that 12% of patients are admitted for surgery, 16% for obstetrics, and 2% for both. What is the probability that a randomly admitted patient is admitted for either surgery or obstetrics or both?

Accepted Answer

$$P(	ext{surgery} \cup 	ext{obstetrics}) = 0.12 + 0.16 - 0.02 = 0.26$$

Question 6

Which of the following statements is TRUE about mutually exclusive events?

Accepted Answer

If events $$A$$ and $$B$$ are mutually exclusive, then $$P(A \cap B) = 0$$

Question 7

Given the following probabilities from a manufacturing process: $$P(DD) = 0.01$$, $$P(DN) = 0.045$$, $$P(ND) = 0.045$$, $$P(NN) = 0.9. $$What is the probability of observing at least one defective smartphone in a sample of two?

Accepted Answer

$$P(	ext{at least one defective}) = 0.01 + 0.045 + 0.045 = 0.1$$

Question 8

Suppose you have 20 business ventures and want to select 5 for investment. How many different samples of 5 ventures can be selected?

Accepted Answer

$$\binom{20}{5} = \frac{20!}{5!\,15!}$$

Question 9

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

Accepted Answer

$$P(A \cap B) = 0.5 	imes 0.4 = 0.2$$

Question 10

A study finds that 55% of executives have cheated at golf, and 20% have both cheated at golf and lied in business. What is the probability that an executive who has cheated at golf has also lied in business?

Accepted Answer

$$P(	ext{lied in business} | 	ext{cheated at golf}) = \frac{0.20}{0.55} = 0.364$$

Question 11

Which of the following is the correct formula for conditional probability?

Accepted Answer

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

Question 12

In a die-toss experiment, let $$A = \{	ext{even number}\}$$ and $$B = \{	ext{number} \leq 3\}. $$What is $$A \cap B$$?

Accepted Answer

$$A \cap B = \{2\}$$

Question 13

Which of the following is NOT a property of probability for sample points?

Accepted Answer

The probability of a sample point can be negative.

Question 14

Bayes’s Rule is used to:

Accepted Answer

Update the probability of an event based on new information.

Question 15

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

Accepted Answer

Events $$A$$ and $$B$$ are independent.

Question 16

A wheelchair user points the joystick straight. Given $$P(D) = 0.5$$, $$P(S) = 0.2$$, $$P(T) = 0.3$$, $$P(J|D) = 0.3$$, $$P(J|S) = 0.4$$, $$P(J|T) = 0.05$$, which destination is most likely?

Accepted Answer

Through the door (event D)

Question 17

Which of the following is the correct formula for the number of combinations of $$n$$ elements selected from $$N$$ elements?

Accepted Answer

$$\binom{N}{n} = \frac{N!}{n!(N-n)!}$$

Question 18

If two events $$A$$ and $$B$$ are independent, which of the following is always true?

Accepted Answer

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

Probability: Foundations and Rules for Business Statistics

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