Mathematics 375 – Probability and Statistics I Midterm Exam 1 – October 7, 2005

Directions: Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points.

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

3.1, 3.5, 3.3, 4.5, 4.2, 2.8, 3.9, 3.5, 3.3, 4.0

A) (10) Construct a relative frequency histogram for this data on the interval [3, 5],

subdividing into 5 equal “bins”. B) (5) What is the sample mean? C) (10) How many of the data points are within 2 standard deviations of the mean?

II. Students on a boat send signals back to shore by arranging (exactly) 9 colored flags on a vertical flagpole. A) (5) How many different signals can they send if they have flags of 9 different colors? B) (10) How many different signals can they send if they have 4 green, 3 red, and 2 blue

flags? (There are no differences between the flags of the same color.) C) (10) In the situation of part B, if a random arrangement of flags is constructed, what

is the probability that all the green flags appear consecutively?

III. Let A_{1}, A_{2}, A_{3} be events in a sample space S. Assume that S = A_{1} ∪ A_{2} ∪ A_{3}, where A_{i} ∩ A_{j} = ∅ if i ≠ j. Let P (A_{1}) = .3, P (A_{2}) = .2, P (A_{3}) = .5. 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).

IV. A) (15) A jar contains 7 red and 11 white balls. Draw two balls at random, without

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 mass function, and compute its expected value and variance.

B) (10) Assume you have a 40% of connecting each time you dial a very busy customer service telephone line. If 25 calls are made at random and independently, what is the probability that between 15 and 20 of them (inclusive) will get through?

Extra Credit (10) Let Y have a geometric distribution with success probability p. Show that the expected value of g(Y) = e^{Y} is E(e^{Y}) = \frac{pe}{1−qe}.