Q: Suppose the probability that a randomly selected worker is female is $$0.17$$, and the probability that a worker is in the plant is $$0.9. If $$the probability that a worker is female and works in the plant is $$0.10$$, are the events 'female' and 'works in the plant' independent?

No, because $$0.10 \neq 0.17 \times 0.9$$

Q: What is the expected value (mean) of the number of calls received in the previous question?

$$0 \times 0.40 + 1 \times 0.20 + 2 \times 0.15 + 3 \times 0.10 + 4 \times 0.10 + 5 \times 0.05 = 1.45$$

Q: You participate in a free-throw challenge. If you make one basket, you win $$50 ($$probability $$0.42$$); if you make two, you win $$500 ($$probability $$0.001$$); otherwise, you win nothing. What is your expected winnings?

$$0 \times 0.579 + 50 \times 0.42 + 500 \times 0.001 = 21.5$$

Q: A supermarket has three employees: A, B, and C. Employee A records the correct weight $$98\% of $$the time and handles $$40\% of $$packages; B records $$97\%$$ and handles $$40\%$$; C records $$95\%$$ and handles $$20\%. If a $$package is found to be incorrectly weighed, what is the probability it was handled by employee C?

$$\frac{0.05 \times 0.20}{0.02 \times 0.40 + 0.03 \times 0.40 + 0.05 \times 0.20}$$

Q: For vehicles on a highway, speeds are normally distributed with mean $$58.4$$ MPH and standard deviation $$5.3$$ MPH. What is the cutoff speed for the slowest $$3\% of $$vehicles?

$$58.4 + (z_{0.03}) \times 5.3$$, where $$z_{0.03} \approx -1.88$$

Question 1

A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{1}{3}$$

Question 2

Two events, A and B, are said to be independent if:

Accepted Answer

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

Question 3

A disk is randomly selected from a batch of 100 disks. 70 have high scratch resistance and high shock resistance, 9 have high scratch resistance and low shock resistance, 16 have low scratch resistance and high shock resistance, and 5 have low scratch resistance and low shock resistance. What is the probability that the disk has high scratch resistance?

Accepted Answer

$$\frac{79}{100}$$

Question 4

In a survey, 42% prefer purple, 37% prefer teal, and 19% prefer pink. What is the probability that a randomly chosen person prefers purple or pink?

Accepted Answer

$$0.61$$

Question 5

If three people are chosen at random from the survey above, what is the probability that at least one prefers purple?

Accepted Answer

$$1 - (1 - 0.42)^3$ = 0.8048$$

Question 6

Which of the following best describes the complement of an event A?

Accepted Answer

All outcomes in the sample space not in A

Question 7

Suppose the probability that a randomly selected worker is female is $$0.17$$, and the probability that a worker is in the plant is $$0.9. If $$the probability that a worker is female and works in the plant is $$0.10$$, are the events 'female' and 'works in the plant' independent?

Accepted Answer

No, because $$0.10 
eq 0.17 	imes 0.9$$

Question 8

A random variable X has the following probability distribution: $$P(X=0)=0.40$$, $$P(X=1)=0.20$$, $$P(X=2)=0.15$$, $$P(X=3)=0.10$$, $$P(X=4)=0.10$$, $$P(X=5)=0.05. $$What is the probability that there will be at least 3 calls in the next 15 minutes?

Accepted Answer

$$0.10 + 0.10 + 0.05 = 0.25$$

Question 9

What is the expected value (mean) of the number of calls received in the previous question?

Accepted Answer

$$0 	imes 0.40 + 1 	imes 0.20 + 2 	imes 0.15 + 3 	imes 0.10 + 4 	imes 0.10 + 5 	imes 0.05 = 1.45$$

Question 10

Sally buys a coffee and a muffin every day. The average price of coffee is $$3.50 ($$standard deviation $$0.30$$), and the average price of a muffin is $$2.50 ($$standard deviation $$0.15). If $$the prices are independent, what is the average amount she spends on breakfast each day?

Accepted Answer

$$3.50 + 2.50 = 6.00$$

Question 11

You participate in a free-throw challenge. If you make one basket, you win $$50 ($$probability $$0.42$$); if you make two, you win $$500 ($$probability $$0.001$$); otherwise, you win nothing. What is your expected winnings?

Accepted Answer

$$0 	imes 0.579 + 50 	imes 0.42 + 500 	imes 0.001 = 21.5$$

Question 12

Which statement about probability distributions is correct?

Accepted Answer

The sum of all probabilities in a probability distribution must equal 1

Question 13

Suppose the heights of male students are normally distributed with mean $$69$$ inches and standard deviation $$3$$ inches. What proportion of students have heights between $$66$$ and $$72$$ inches?

Accepted Answer

Approximately $$95\%$$

Question 14

For a normal distribution, what is the probability that a randomly selected value falls within one standard deviation of the mean?

Accepted Answer

About $$68\%$$

Question 15

A supermarket has three employees: A, B, and C. Employee A records the correct weight $$98\% of $$the time and handles $$40\% of $$packages; B records $$97\%$$ and handles $$40\%$$; C records $$95\%$$ and handles $$20\%. If a $$package is found to be incorrectly weighed, what is the probability it was handled by employee C?

Accepted Answer

$$\frac{0.05 	imes 0.20}{0.02 	imes 0.40 + 0.03 	imes 0.40 + 0.05 	imes 0.20}$$

Question 16

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.5 = 0.8$$

Question 17

For vehicles on a highway, speeds are normally distributed with mean $$58.4$$ MPH and standard deviation $$5.3$$ MPH. What is the cutoff speed for the slowest $$3\% of $$vehicles?

Accepted Answer

$$58.4 + (z_{0.03}) 	imes 5.3$$, where $$z_{0.03} \approx -1.88$$

Question 18

Which formula represents the Law of Total Probability for two mutually exclusive and exhaustive events $$B$$ and $$B^c$$?

Accepted Answer

$$P(A) = P(B)P(A|B) + P(B^c)P(A|B^c)$$

Probability and Probability Distributions: Core Concepts for Introductory Statistics

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