Q: A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported on a given day are 0.53, 0.43, 0.03, and 0.01, respectively. What is the expected number of burglaries reported on a given day?

$$0 \times 0.53 + 1 \times 0.43 + 2 \times 0.03 + 3 \times 0.01 = 0.55$$

Q: If a random variable X has the following probability distribution: $$P(X=0)=0.5$$, $$P(X=1)=0.3$$, $$P(X=2)=0.2$$, what is the expected value $$E[X]$$?

$$0 \times 0.5 + 1 \times 0.3 + 2 \times 0.2 = 0.7$$

Q: If the probability that a randomly selected student does not have a Facebook account is 0.48, and the probability that the student is female is 0.52, what is the probability that a randomly selected student is either female or does not have a Facebook account, assuming independence?

$$0.52 + 0.48 - (0.52 \times 0.48) = 0.7488$$

Question 1

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

Accepted Answer

The set of all possible outcomes of a random experiment.

Question 2

Two events A and B are said to be disjoint if:

Accepted Answer

They cannot occur at the same time.

Question 3

Suppose the probability that a randomly selected student has health insurance is 0.6, and the probability that the student has car insurance is 0.7. If the probability that a student has both is 0.4, what is the probability that a student has either health or car insurance?

Accepted Answer

$$0.6 + 0.7 - 0.4 = 0.9$$

Question 4

If a fair coin is tossed 5 times, what is the probability of getting exactly 3 heads?

Accepted Answer

$$\frac{10}{32} = 0.3125$$

Question 5

A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported on a given day are 0.53, 0.43, 0.03, and 0.01, respectively. What is the expected number of burglaries reported on a given day?

Accepted Answer

$$0 	imes 0.53 + 1 	imes 0.43 + 2 	imes 0.03 + 3 	imes 0.01 = 0.55$$

Question 6

Which of the following is the correct formula for the probability of the union of two events A and B?

Accepted Answer

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

Question 7

A manufacturer finds that 2% of its chips are defective. If an inspector randomly selects 5 chips, what is the probability that at least one chip is defective? (Assume independence.)

Accepted Answer

$$1 - (0.98)^5$ \approx 0.0961$$

Question 8

If a random variable X has the following probability distribution: $$P(X=0)=0.5$$, $$P(X=1)=0.3$$, $$P(X=2)=0.2$$, what is the expected value $$E[X]$$?

Accepted Answer

$$0 	imes 0.5 + 1 	imes 0.3 + 2 	imes 0.2 = 0.7$$

Question 9

A student selects 5 novels from a reading list of 20 for a course in literature. How many different possible ways can this selection be made?

Accepted Answer

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

Question 10

Suppose the heights of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy lasts at least 300 days?

Accepted Answer

$$P(Z \geq \frac{300-268}{15}) = P(Z \geq 2.13)$$

Question 11

Which of the following is NOT a requirement for a binomial probability experiment?

Accepted Answer

The trials must be dependent.

Question 12

If $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.3$$, what is $$P(A \mid B)$$?

Accepted Answer

$$\frac{P(A \cap B)}{P(B)} = \frac{0.3}{0.5} = 0.6$$

Question 13

A life insurance company charges $185.85 for a male who will live through the year. If the probability that the male will not survive the year is 0.002, what is the expected payout by the company?

Accepted Answer

$$0.002 	imes 100,000 = 200$$

Question 14

If the probability that a randomly selected student does not have a Facebook account is 0.48, and the probability that the student is female is 0.52, what is the probability that a randomly selected student is either female or does not have a Facebook account, assuming independence?

Accepted Answer

$$0.52 + 0.48 - (0.52 	imes 0.48) = 0.7488$$

Question 15

Which of the following best describes a random variable?

Accepted Answer

A function that assigns a real number to each outcome in the sample space.

Question 16

If the standard deviation of a normal distribution is 15 and the mean is 100, what is the z-score for a value of 130?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 17

A deck consists of 5 letters followed by 2 digits. How many cards can be made if repetitions are allowed?

Accepted Answer

$$26^5$$ 	imes $$10^2$$

Question 18

Which of the following is the formula for the expected value $$E[X] of a $$discrete random variable X?

Accepted Answer

$$E[X] = \sum x_i p_i$$

Question 19

If the probability that a randomly selected adult did not visit their physician's office last year is 0.18, what is the expected number of adults out of 50 who did not visit their physician's office?

Accepted Answer

$$50 	imes 0.18 = 9$$

Probability and Statistics: Key Concepts, Definitions, and Problem-Solving Strategies

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