Claudia Pena
September 30, 2023
BUS219
Chapter 4: INTRODUCTION TO PROBABILITY
1.
Determine whether the following probabilities are best categorized as subjective, empirical, or classical probabilities.
a)
Before flipping a fair coin, Sunil assesses that he has a 50% chance of obtaining tails.
·
b)
At the beginning of the semester, John believes he has a 90% chance of receiving straight A’s.
·
c)
A political reporter announces that there is a 40% chance that the next person to come out of the conference room will be a Republican, since there are 60 Republicans and 90 Democrats in the room.
·
5.
Jane Peterson has taken Amtrak to travel from New York to Washington, DC, on six occasions, of which three times the train was late. Therefore, Jane tells her friends that the probability that this train will arrive on time is 0.50.
Would you label this probability as empirical or classical? Why would this probability not be accurate?
·
7.
Consider the following scenarios to determine if the mentioned combination of attributes represents a union or an intersection.
a)
A marketing firm is looking for a candidate with a business degree and at least five years of work experience.
·
b)
A family has decided to purchase a Toyota minivan or a Honda minivan.
·
9. An alarming number of U.S. adults are either overweight or obese. The distinction between overweight and obese is made on the basis of body mass index (BMI), expressed as weight/height2. An adult is considered overweight if the BMI is 25 or more but less than 30. An obese adult will have a BMI of 30 or greater. A recent study suggests that 33.1% of the adult population in the United States is overweight and 35.7% is obese. Use this information to answer the following questions.
a)
What is the probability that a randomly selected adult is either overweight or obese?
·
b)
What is the probability that a randomly selected adult is neither overweight nor obese?
·
c)
Are the events “overweight” and “obese” exhaustive?
·
d)
Are the events “overweight” and “obese” mutually exclusive?
·
10. At four community health centers on Cape Cod, Massachusetts, 15,164 patients were asked to respond to questions designed to detect depression. The survey produced the following results.
|
Diagnosis |
Number |
|
Mild |
3,257 |
|
Moderate |
1,546 |
|
Moderately Severe |
975 |
|
Severe |
773 |
|
No Depression |
8,613 |
a)
What is the probability that a randomly selected patient suffered from mild depression?
·
b)
What is the probability that a randomly selected patient did not suffer from depression?
·
c)
What is the probability that a randomly selected patient suffered from moderately severe to severe depression?
·
d)
Given that the national figure for moderately severe to severe depression is approximately 6.7%, does it appear that there is a higher rate of depression in this summer resort community? Explain.
·
Section 4.2
11. Let P(A) = 0.65, P(B) = 0.30, and P(A|B) = 0.45.
a)
Calculate P(A ∩ B).
–
b)
Calculate P(A ∪ B).
·
c)
Calculate P(B|A).
·
12. Let P(A) = 0.55, P(B) = 0.30, and P(A ∩ B) = 0.10.
a)
Calculate P(A|B).
·
b)
Calculate P(A ∪ B).
·
c)
Calculate P((A ∪ B)c).
·
18. Only 20% of students in a college ever go to their professor during office hours. Of those who go, 30% seek minor clarification and 70% seek major clarification.
a)
What is the probability that a student goes to the professor during her office hours for a minor clarification?
·
b)
What is the probability that a student goes to the professor during her office hours for a major clarification?
·
19. The probabilities that stock A will rise in price is 0.40 and that stock B will rise in price is 0.60. Further, if stock B rises in price, the probability that stock A will also rise in price is 0.50.
a)
What is the probability that at least one of the stocks will rise in price?
·
b)
Are events A and B mutually exclusive? Explain.
·
c)
Are events A and B independent? Explain.
·
20. Fraud detection has become an indispensable tool for banks and credit card companies to combat fraudulent credit card transactions. A fraud detection firm raises an alarm on 5% of all transactions and on 80% of fraudulent transactions.
What is the probability that the transaction is fraudulent if the firm does not raise an alarm? Assume that 1% of all transactions are fraudulent.
·
29. Christine has asked Dave and Mike to help her move into a new apartment on Sunday morning. She has asked them both, in case one of them does not show up. From past experience, Christine knows that there is a 40% chance that Dave will not show up and a 30% chance that Mike will not show up. Dave and Mike do not know each other and their decisions can be assumed to be independent.
a)
What is the probability that both Dave and Mike will show up?
·
b)
What is the probability that at least one of them will show up?
·
c)
What is the probability that neither Dave nor Mike will show up?
·
Section 4.3:
33. Consider the following contingency table.
a)
Convert the contingency table into a joint probability table.
b)
What is the probability that A occurs?
·
c)
What is the probability that A and B occur?
·
d)
Given that B has occurred, what is the probability that A occurs?
·
e)
Given that Ac has occurred, what is the probability that B occurs?
·
f)
Are A and B mutually exclusive events? Explain.
·
g)
Are A and B independent events? Explain.
·
38. (FILE) There have been numerous attempts that relate happiness with income. In a recent survey, 290 individuals were asked to evaluate happiness (Yes or No) and income (Low, Medium, or High). Below are the evaluations, zoom in the picture to see the information more lear.
The accompanying table shows a portion of the data.
|
Individual |
Income |
Happy? |
|
1 |
Low |
No |
|
2 |
Low |
Yes |
|
⋮ |
⋮ |
⋮ |
|
290 |
High |
Yes |
a)
Use the data to construct a contingency table.
·
b)
Find the probability that a randomly selected individual feels happy.
·
c)
Find the probability that a low-income individual feels happy. Find the corresponding probabilities for medium-income and high-income individuals.
·
d)
Is income related to happiness? Explain using probabilities.
·
Section 4.5
51. Calculate the following values.
a)
8! and 6!
·
b)
8C6
·
c)
8P6
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53. Twenty cancer patients volunteer for a clinical trial. Ten of the patients will receive a placebo and 10 will receive the trial drug. In how many different ways can the researchers select 10 patients to receive the trial drug from the total of 20?
·
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