Probability and Statistical Inference Exercises

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Students are to complete the exercises in Word (or some other compatible word processor) and submit.

1. What is wrong with the following statements? Please explain. 

a. P(A) = 2.3
b. P(complement of A) = 0.7 given that P(A) = 0.7
c. P(A or B) = 1.0 if P(A) = 0.2 and P(B) = 0.5
d. P(A and B) = -.2

2. Given that P(A) = 0.42, P(B) = 0.50 and P(A and B) = 0.20, calculate the following: 

a. P(A or B)

b. P(the complement of B)

c. P(the complement of the event “A or B”)
d. P(A|B)
e. P(B|A)

3. The following table is a summary of one of the key questions in the employee satisfaction survey for ABC Limited. If an employee (a male) is randomly
selected, answer the questions below: (7 points)

Do you plan to be with Company one year from now?

Downtown        40           4
Uptown          92           8

a. What is the probability that he is from the Uptown branch?

b. What is the probability that he plans to be with the company one year from now?

c. What is the probability that he is from the Downtown branch and plan to be with the company one year from now?

d. What is the probability that he is from the Uptown branch or does not plan to be with the company one year from now?

e. What is the probability that he is from the Downtown branch if he plans to be with the company one year from now?

f. What is the probability that he plans to be with the company one year from now given that he is from the Uptown branch?

g. Are the events “Downtown Branch” and “plans to be with the company one year from now” independent? Justify your answer using probabilities.

4. Historically, 45% of visitors to the online store of XYZ company made a purchase. 40% of the visitors who made a purchase came to the online store through a web search engine. Also, 15% of visitors who did not make a purchase came to the online store through a web search engine. If a visitor comes to the online store through a web search engine, what is the probability that he or she will make a purchase?

(Suggestion: To think through the problem, use a probability tree and “whether a visitor makes a purchase” as the first step to depict what is given. Use the Bayes Theorem to calculate the probability in question. You do not need to include the tree in your submission.).

5. a. Find the value of W.
Probability (f(X))
0.15
0.20
0.45
W

b. What is the probability that the company has a net income of less than $1 million in the coming year?

c. What is the probability that the company is not profitable (net income leas than 0) in the coming year?

d. What is the probability that the company makes more than $0.5 million in the coming year?

e. What is the expected net income for ABC Limited, i.e, what is the expected value of this probability distribution?

f. What is the standard deviation of this probability function?

6. a. Find the value of W.

Probability (f(X))
0.15
0.20
0.45
W

b. What is the probability that the company has a net income of less than $1 million in the coming year?

c. What is the probability that the company is not profitable (net income leas than 0) in the coming year?

d. What is the probability that the company makes more than $0.5 million in the coming year?

e. What is the expected net income for ABC Limited, i.e, what is the expected value of this probability distribution?

f. What is the standard deviation of this probability function?

7. Customers arrive at the drive-through lane of a fast food restaurant at a rate of one every 3 minutes. Use the Poisson probability distribution to answer the following questions: 

a. What is the expected number of calls in one hour?

b. What is the probability that exactly two customers arriving at the drive- through lane in a nine-minutes interval?

c. What is the probability that less than two customers arrive at the drive- through lane in a nine-minutes interval?

d. What is the probability that two or more customers arrive at the drive- through lane in a nine-minutes interval?