**Probability and Counting Rules **

The relevant R codes and outputs must be attached for full credit.

1. Explain whether or not the following numbers could be examples of a probability.

**(8 points total 2 points each)**

a) *P*(A) = 0.5

b) *P*(*B*) = 0

c) *P*(*C*) = 1.6

d) *P*(*D*) = -3

2. A quality control engineer randomly selects two light bulbs from a shipment to determine if they meet all the specifications defined by a company. Let M = a light bulb meets all the specifications defined by a company, and N = a light bulb does NOT meet all the specifications defined by a company. Write the sample space of this experiment in set notation. **(1 point) **

3. Consider *x* be an event defined by the waiting time (in hours) between successive speeders spotted by a radar unit. Write the sample space of this event using mathematical notation*. ***(1 point)**

4. Consider the sample space *S* = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events

A = {copper}

B = {sodium, nitrogen, potassium}

C = {oxygen}

List the elements of the sets corresponding to the following events *using set notation*:

Source: Walpole, R.E., Myers, R.H., Myers, S.L., & Ye, K. (2012). *Probability & Statistics for Engineers & Scientists*. Boston, MA: Pearson.

a. BC **(1 point) **

b. A B C **(1 point) **

c. (A B)C (AC C) **(1 point) **

d. B C **(1 point)**

5. A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma? **(1 point)**

6. A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer? **(1 point)**

7. How many distinct ways are there to arrange the letters of the word “Miami”? (Note: Assume capital M and lowercase m are the same letter) **(1 point)**

**Note: **Students can use RStudio or a formula.

8. A statistics class for engineers consists of 53 students. The students in the class are classified based on their college major and sex as shown in the following contingency table:

College Major

Sex

Industrial Engineering

Mechanical Engineering

Electrical Engineering

Civil Engineering

Total

Male

15

6

7

2

30

Female

10

4

3

6

23

Total

25

10

10

8

53

If a student is selected at random from the class by the instructor to answer a question, find the following probabilities. Report your answer to 4 decimal places.

Consider the following events:

A: The selected student is a male.

B: The selected student is industrial engineering major.

C: The selected student is electrical engineering major.

D: The selected student is civil engineering major.

**Note: **Indicate the type of probability as marginal, joint or conditional when asked.

a) Find the probability that the randomly selected student is a male. Indicate the type of probability. **(1 + 1 = 2 points)**

b) Find the probability that the randomly selected student is industrial engineering major. Indicate the type of probability. **(1 + 1 = 2 points)**

c) Find the probability that the randomly selected student is male industrial engineering major. Indicate the type of probability. **(1 + 1 = 2 points)**

d) Given that the selected student is industrial engineering major, what is the probability that the student is male? Indicate the type of probability.

**(1 + 1 = 2 points)**

e) Based on your answers on part a and d, are sex and college major of students in this class independent? Provide a mathematical argument? **(1 point)**

f) Consider the events A and B. Are sex and college major mutually exclusive events? Provide a mathematical argument to justify your answer. **(1 point)**

g) Find the probability that the randomly selected student is male or industrial engineering college major. **(1 point)**

h) Consider the events C and D. Are college major mutually exclusive events? Provide a mathematical argument to justify your answer. **(1 point)**

i) Find the probability that the randomly selected student is electrical or civil engineering college major. **(1 point)**

j) What is the probability that a randomly selected student is neither a male nor an industrial engineering college major **(1 point)**

9. Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations (location 1, location 2, location 3, and location 4, respectively) will be operated 40%, 30%, 20%, and 30% of the time. A person who is speeding on his/her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations (location 1, location 2, location 3, and location 4, respectively).

a) If a speeding person is randomly selected, what is the probability that he/she will receive a speeding ticket? Report your answer to 4 decimal places. **(5 points)**

**Hint:** Defining events and making a tree diagram would be very helpful. Students may use the following template to produce a tree diagram.

**Tree Diagram**

b) If this person received a speeding ticket on his/her way to work, what is the probability that he/she passed through the radar trap located at location 2?

**(2 points)**

10. Assume a small town has two fire engines operating independently. The probability that a specific engine is available when needed is 0.94.

a) Find the probability that both fine engines are available when needed. Report your answer to four decimal places. **(1 point)**

b) What is the probability that neither fire engine is available when needed? Report your answer to four decimal places. **(1 point)**

c) What is the probability that at least one fire engine is available when needed)? Report your answer to four decimal places. **(2 points)**

11. The probability that a doctor correctly diagnoses a particular illness is 0.75. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.85. What is the probability that the doctor makes an incorrect diagnosis and the patient sues? Report your answer to four decimal places. **(2 points)**

12. The probability that a patient survives from a delicate heart operation is 0.8.

a) What is the probability that exactly 2 of the next 3 patients who have this operation survive? Report your answer to four decimal places. **(2 points)**

b) What is the probability that all of the next 3 patients who have this operation survive? Report your answer to four decimal places. **(2 points)**

c) What is the probability that all of the next 3 patients who have this operation die? Report your answer to four decimal places. **(2 points)**