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Math 29 – Probability

Practice First Midterm Exam 2

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 2 and 3 share a page.)
  5. Good luck!

Problem / 1 / 2 / 3 / 4 / 5 / Total
Points Earned
Possible Points / 13 / 6 / 8 / 12 / 11 / 50

1. A town nicknamed “Mostly Cloudy and Depressing” has the following distribution for daily weather in the summer:

Sunny / Partly Cloudy / Cloudy / Raining
.15 / .25 / .35

It never snows in the summer and there is never any weather other than shown. Also, unique to this town, each day’s weather is independent of any other day’s weather.

a. What must P(Raining) be? Complete the table above.

b. Suppose the town’s 10 day summer fair can only be held if the weather is sunny or partly cloudy. What distribution does X, the number of days in 10 that the 10 day fair operates, have?

c. What is the probability that the fair only occurs two days out of the 10?

d. What is the probability the fair occurs on 8 or fewer days out of the 10?

e. Now suppose the fair decides to try to operate all summer (90 days). Assume the initial setup cost is $2000. Additionally, every day, the cost of having the fair ready to operate is $150. Each day that the fair is in operation generates revenue of $800. What is the expected profit for the fair in the summer as well as the standard deviation of profit?

RST + Result / RST - Result / Total
Strep (+) / 159 / 36
No Strep (-) / 3 / 1302
Total

2. A new test for strep throat called the Rapid Strep Test (RST) underwent testing and is now used (along with traditional strep tests) to test for strep throat quickly while in a doctor’s office. Here are some test results related to the RST testing:

a. What is the sensitivity of the RST?

b. What is the specificity of the RST?

c. What is the predictive value of the RST?

3. The commander of a military operation has 30 active soldiers under his command, and needs to select 10 for a mission. Of the 30 soldiers, 15 are recon specialists, 10 are snipers, and the rest are survival specialists. (You do not need to compute the values out for the parts below.)

a. In how many ways can the commander select 10 soldiers for the mission?

b. In how many ways can a team be constructed so that it has just one survival specialist?

c. Suppose the optimal desired makeup is 6 recon specialists, 3 snipers, and 1 survival specialist. How many possible teams have this optimal makeup?

d. After the mission, the 10 soldiers return to a waiting pile of generic care packages which come in 3 different sizes. Suppose there are 3 large, 3 medium, and 4 small care packages available. Packages of the same size are indistinguishable. In how many ways can the packages be distributed to the 10 soldiers (1 per soldier)?

4. A machine produces items that are rated as high, medium, or poor in quality. If the machine is properly adjusted, it produces 40% high quality items, 50% medium quality items, and the rest poor quality items. If the machine is improperly adjusted, it produces only 20% high quality items, 40% medium quality items, and the rest poor quality items. Assume that the machine is properly adjusted 80% of the time. Items should be treated as independent for purposes of rating their quality.

a. If you sample one item and it is of medium quality, what is the probability the machine is properly adjusted at that time?

b. If you sample one item and it is of poor quality, what is the probability that the machine is improperly adjusted at that time?

c. Suppose you sample 3 items in a short time span (it cannot shift adjustment level during that time) and 2 items are medium quality and one is poor quality. What is the probability that the machine is improperly adjusted at that time?

5. Suppose X is a general discrete random variable taking on the values x=1, 2, 3, 4 and 5, with p(x)=cx.

a. What value of c makes this a valid pmf? (Show work to justify your answer.)

b. What are the mean and variance of X?

c. Using Tchebysheff’s Theorem, provide an interval of values within which at least 75% of the distribution of X should lie.

d. Give an interval containing at least 75% of the values in the distribution of X without using Tchebysheff’s Theorem.

e. A coin with both sides being heads, so that all the probability mass is concentrated on a single value (heads) gives rise to a distribution which is called ______.