Mathematics 375 – Probability Theory Midterm Exam 1 – October 7, 2011

Directions: Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points. Note: If an answer can be expressed in terms of factorials, binomial coefficients, multinomial coefficients etc., please leave it in that form rather than evaluating it further.

I. (15) To test the effectiveness of a seal designed to keep a tire patch airtight, a needle was inserted into the tire and air pressure was increased until leakage was observed. The pressures (in lb per square inch) where leakage first occurred on 10 trials were:

8.1, 8.5, 8.3, 8.8, 8.2, 7.8, 8.9, 7.5, 8.3, 8.0

How many of the data points are within 1 standard deviation of the mean? How well does this fit the “empirical rule?”

II. Students on a boat send signals back to the shore by stringing (exactly) 9 colored flags along a vertical rope (without overlapping). The signal is determined solely by the relative positions of the flags. A) (5) How many different signals can they send if they have flags of 9 different colors? B) (10) Now suppose they have 4 red flags and 5 green flags. If a random arrange-

ment of flags is constructed, what is the probability that all the green flags appear consecutively?

III. The first three parts of this question deal with this situation: Let A_{1}, A_{2}, A_{3} be events in a sample space S. Let P (A_{1}) = .4, P (A_{2}) = .2, P (A_{3}) = .4, and assume P (A_{i} ∩A_{j}) = ∅ if i ≠ j. Finally, let B be another event with P (B|A_{1}) = .1, P (B|A_{2}) = .2 and P (B|A_{3}) = .05. A) (10) What is P (B)? B) (5) Are B and A_{2} independent events? Why or why not? C) (10) Find P (A_{1} ∪ A_{2}|B). D) (15) Now, working in general, state and prove Bayes’ Rule.

IV. A) (20) A jar contains 7 red and 11 white balls. Draw two balls at random, replacing the

first ball after it is drawn. Let X be the number of balls drawn that are red. Thinking of X as a discrete random variable, find its probability function, and compute its expected value and variance.

B) (10) Now consider drawing two balls without replacing the first ball after it is drawn, and let Z be the number of red balls drawn. Give the probability function for this Z.

Extra Credit (10)

The probability that a randomly chosen male has a circulation problem is .25. Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem. What is the probability that a male has a circulation problem, given that he is a smoker?