Mathematics 375 – Probability and Statistics 1 Exam 1 – October 10, 2003

Directions Do all work in the blue exam booklet. There are 100 regular points and 10 extra credit points.

I. (20) A manufacturer of electronic components tests the lifetimes of a certain type of battery and finds the following data:

123, 116, 122, 110, 125, 126, 111, 118, 117, 120

(lifetimes in hours). How many of the sample points are within one standard deviation of the sample mean? Is there reason to believe the lifetime of this type of battery is not normally distributed from this small sample? Explain.

II. (15) In a regional spelling bee, the 10 finalists consist of 5 girls and 5 boys. Assume that all the finalists are equally proficient spellers and that the outcome of the contest is random. What is the probability that 4 of the top 5 finishers will be female?

III. A) (15) State and prove the Law of Total Probability. B) (10) The Podunk City police department plans a crackdown on speeders by placing

radar traps at four different locations L_{1}, L_{2}, L_{3}, L_{4}. The probability that each of the traps is manned at any one time is .4, .3, .2, .3 respectively (and the police really mean business – if a trap is manned every speeder who passes it will get a ticket). Speeders have probabilities of passing the four locations of .2, .1, .5, .2 respectively, and no one passes more than one. What is the probability that a given speeder will actually receive a ticket?

C) (10) In the situation of part B, given that a speeder received a ticket, what is the probability that he passed location L_{2}?

IV. (10) Let A, B be events for which P (A) = .2, P (B) = .3 and P (A ∩ B) = .06. Are \overline{A} and \overline{B} independent events?

V. An allergist knows that 30% of all people are allergic to the pollen of burdock weed. A) (10) Starting from some point in time, the first five patients in sequence that the

allergist sees are not allergic to burdock pollen. Given that, what is the probability that the first patient the doctor sees who is allergic to burdock pollen will come after the 20th patient seen? Explain the assumptions you are making to derive your solution.

B) (10) What is the probability that from 4 to 7 (inclusive) of the next 20 patients she sees are allergic to burdock pollen? Explain the assumptions you are making to derive your solution.

(Turn over for Extra Credit problem.)

Extra Credit (10) The Yankees won 10 of the 19 regular season games they played against the Red Sox this year. Assuming that the Yankees have a 10/19 chance of winning each game they play against the Sox, as of last Tuesday (i.e. before the first game), what was the probability that the Sox will clinch the ALCS at home? Assume the games are independent events, and recall that the series starts with two games in New York, then moves to Boston for the next 3 (if needed), then back to New York for the remainder of the series (if needed).

Go Sox! Cowboy Up!