STA 4321 / 5325

Sample exam 1February 18, 2015

Name:

UFID:

This is a 50-minute exam. There are 4 problems, worth a total of 40 points. Remember

to show your work. Answers lacking adequate justification may not receive full credit.

You may use one 8.5 by 11 sheet with formulas or notes on both sizes and a pocket calcu-

lator. You may not use any books, other references, or text-capable electronic devices.

Problem 1. A company buys microchips from three suppliers. Supplier I microchips have

8% chance of being defective, Supplier II microchips have 10% chance of being defective

and Supplier III microchips have 6% chance of being defective. Suppose 30%, 35% and

35% of the current supply came from Suppliers I, II and III respectively.

a. If a microchip is selected randomly from this supply, what is the probability that it is

defective? (5 pts)

b. if a randomly selected microchip is defective, what is the probability that it come from

Supplier II? (5 pts)

Problem 2. An automated phone call routing system serves three offices, numbered 1, 2,

and 3. Three phone calls come in, one intended for each of the three offices. Unfortunately,

the phone system is malfunctioning – it assigns one call to each office, but in a random

order. Assume all such assignments are equally likely.

a. Define labels for the outcomes of this experiment. Using these labels for sample points,

list the sample space. (4 pts)

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b. Define the events

A = {office 1 receives its correct call}

B = {office 3 receives its correct call}

C = {no office receives its correct call}

List A, B, and C in terms of the sample space defined above. Which pairs of these events

are disjoint? (4 pts)

c. Are A and B independent events? Why or why not? (Show your work.) (2 pts)

Problem 3. A fisherman is restricted to catching at most three red grouper per day when

fishing in the Gulf of Mexico. A field agent for the wildlife commission often inspects the

day’s catch for boats as they come to shore near his base. He has found that the number of

red grouper caught is 0 with probability 0.1, 1 with probability 0.1, 2 with probability 0.7,

and 3 with probability 0.1. Suppose that these records are representative of the red grouper

daily catches in the Gulf. Let

X = Daily catch of red grouper in the Gulf for a single fisherman.

a. Find E(X) and V (X). (5 pts)

b. Find E(X3 −X2). (5 pts)

Problem 4. In tests of stopping distance for automobiles, cars traveling 30 miles per hour

before the brakes were applied tended to travel distances that appeared to be uniformly

distributed between two points a and b. Find the probabilities of the following events.

a. One of these automobiles, selected at random, stops closer to a than to b. (2 pts)

b. One of these automobiles, selected at random, stops at a point where the distance to a is

more than 9 times the distance to b. (4 pts)

c. Suppose that two automobiles (which behave independently) are used in the test. Find

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the probability that exactly one of the two travels past the midpoint between a and b. (4 pts)

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