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Math 29 – Probability

Practice First Midterm Exam 1

Instructions:

1. Show all work. You may receive partial credit for partially completed problems.

2. You may use calculators and a one-sided sheet of reference notes. You may not use any other references or any texts.

3. You may not discuss the exam with anyone but me.

4. Suggestion: Read all questions before beginning and complete the ones you know best first. Point values per problem are displayed below if that helps you allocate your time among problems. (Problem 3 and 4 share a page.)

5. Good luck!

Problem / 1 / 2 / 3 / 4 / 5 / TotalPoints Earned

Possible Points / 12 / 13 / 5 / 10 / 10 / 50

1. A manufacturing company has come under fire lately for the reliability of its products. Assume for a particular type of office chair, the chairs coming off the assembly line have been classified as perfectly fine, defective, or seriously defective. Furthermore, assume there are 3 different assembly line managers responsible for maintaining the production over the three shifts at the factory. Let X and Y be random variables defined as follows with distribution specified in the table below:

X = 1, perfectly fine chair (PF) Y = 1, morning shift manager (M)

2, defective chair (D) 2, afternoon shift manager (A)

3, seriously defective chair (SD) 3, overnight shift manager (E)

X\Y / 1 (M) / 2 (A) / 3 (E)1 (PF) / .15 / .42 / .23

2 (D) / .04 / .05 / .06

3 (SD) / .01 / .03 / .01

a. What proportion of chairs produced by the company are perfectly fine?

b. What proportion of chairs produced by the company are made under the supervision of the afternoon shift manager?

c. Are perfectly fine status (event PF) and overnight shift status (event E) independent? Justify your answer.

d. A company executive sees the table and decides that the afternoon shift manager should be fired because .05+.03 = .08 is larger than .04+.01=.05 and .06+.01=.07. Using probability, provide a rational explanation for why the afternoon shift manager should NOT be fired (and why the company might consider finding another executive).

2. Students selling protein bars as part of a fundraiser are going door-to-door to faculty offices. At each office, either a single protein bar is sold (probability .3) or no protein bar is sold (faculty member either doesn't want one or isn't present, etc.). The students have decided that they need to sell 10 protein bars before stopping for the day.

a. What is the probability that the students sell their first protein bar at the fifth faculty office?

b. If the students have gone to 3 offices and not yet sold a protein bar, what is the probability they sell their first protein bar at the seventh faculty office?

c. (Setup, but do NOT evaluate). What is the probability that the students have sold their 10 protein bars for the day by visiting 45 or fewer offices?

d. What is the expected value and standard deviation for the number of offices the students have to visit to sell their 10 protein bars for the day?

3. An ice cream company has three suppliers of their dairy based ingredients. Each week, they purchase those ingredients from one supplier. 20% of the time they buy from supplier A, 50% from supplier B, and the rest of the time from supplier C. Recent scares of chemical (melamine) contamination have required checks on the supply. Suppose that 50% of A’s recent shipments were contaminated, as well as 10% of B’s and 20% of C’s. If an ice cream batch one week ended up contaminated, what is the probability that the ice cream company bought the ingredients from supplier A that week?

4. At a large university, there are 20 archaeology graduate students. A professor has obtained the appropriate permissions for a dig and plans to take three to six students on this field expedition depending on funding. (You do not need to compute the values out for the parts below.)

a. How many possible expeditions are there?

b. The professor has great news for the students. The expedition has been fully funded, so six students will be going. Now how many possible expeditions are there?

c. The three most senior graduate students hope to be part of the expedition together. What is the probability the three of them end up together on the fully funded expedition assuming all students are equally likely to be picked to go?

d. If the fully funded expedition requires the students to be divided into a group of three research assistants and a group of three field assistants, how does that affect the number of possible expeditions? (i.e. what is the new number of possible expeditions?)

x / 0 / 1 / 2 / 3 / 4 / 5p(x) / .1 / .2 / .3 / .2 / .05

5. Suppose you arrive at your local banking establishment and that the distribution of X, the number of customers currently in line upon your arrival is given at right. Use the distribution provided to answer the following questions.

a. Complete the provided pmf so that it is a valid pmf.

b. Determine and state the cdf of X, and then sketch it, being sure to satisfy the properties of cdfs.

c. Assume one day you are fairly impatient and decide not to stay if 3 or more people are waiting in line when you arrive. What is the probability you stay?

d. If it takes exactly three minutes for each person to be waited on and leave, and you decide to stay upon arrival no matter how many people are waiting, how long do you expect to wait on average? What is the standard deviation of how long you wait?