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Homework #4. Due: Wednesday, September 22, 1999.
IE 230
Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for Engineers,
John Wiley & Sons, New York, 1999. Chapter 3, Sections 3.6-3.8. Page 4 of the concise
notes.
Throughout, use the step-by-step approach whenever possible.
1. (From Problem 3-74.) Eight cavities in an injection-molding tool produce plastic connectors
that fall into a common stream. A random connector is sampled from the stream every
several minutes. Suppose that five connectors are to be gathered. Let Ai denote the event
that the ith connector is from the first cavity, i = 1, 2,..., 5. Assume that these events are
independent and that the cavities are equally likely.
(a) Write the notation for the event that all five connectors are from the first cavity.
(b) Find the probability of the event of Part (a).
(c) What is the probability that the five connectors were all produced in the same cavity?
(d) What is the probability that exactly four out of the five connectors came from the first
cavity?
2. Consider the random experiment whose procedure is to throw one four-sided die. Let the
sample space be S = {1, 2,..., 4}, where the number indicates the side facing down.
Assume that all four sides are equally likely. Let E2 = {1, 2}, E3 = {1, 3}, and E4 = {1, 4}.
(a) Show that E2 and E3 are independent.
(b) Show that three events E2 , E3 , and E4 are pairwise independent.
(c) Use the definition to show that the three events E2 , E3 , and E4 are not jointly
independent.
(d) Compute P(E4 | E2 E3 ). Explain in words why your answer makes sense.
(e) Roughly, event independence means that knowing whether an event occurs is not
useful for predicting whether another event occurs (in the same replication of the
experiment). Therefore, because of pairwise independence, E4 helps to predict neither E2
nor E3 . Because the three events are not jointly independent, something can be predicted.
What?
3. (From Problem 3-109.) If the events A and B are independent, prove that A' and B' are
independent. (To think about, but not to turn in: If A and B are independent, can A and B'
be independent? Must they be?)
4. (From Problem 3-76.) A random circuit, composed of a network of n devices, operates if and
only if there is a path from beginning to end. The path cannot pass through a device that
is not working. Let Di denote the event that device i works, i = 1, 2,..., n. Assume that
these events are jointly independent.
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Homework #4. Due: Wednesday, September 22, 1999.
IE 230
Consider the network of six devices structured with a series of three pairs of parallel devices, as
illustrated in the textbook.
(a) Sketch the network as shown in the textbook, indicating how you are numbering the
devices.
(b) Using Di event notation, write the probabilities given in the textbook.
(c) Write, in set notation, the event that the network operates.
(d) Compute the probability of the event in Part (c).
5. (From Problem 3-80.) Software to detect fraud in consumer phone cards tracks the number of
metropolitan areas where calls originate each day. One percent of legitimate users
originate calls from two or more areas in a single day. Thirty percent of fraudulent users
originate calls from two or more areas in a single day. The proportion of fraudulent users
is 0.01%. Suppose that we randomly select a user who has made calls from two or more
areas in a single day. We are interested in the probability that such a user is fraudulent.
(a) Define event notation for the user being fraudulent and for the user making calls from
more than one area in a day.
(b) Write what you know in terms of these events.
(c) Are the events of Part (a) independent? Why or why not?
(d) Determine the probability of interest.
6. Spread sheet. (Email a copy to "[email protected]". Hand in a hard copy, but separate
from Problems 1-5.)
We will perform a Monte Carlo simulation, at first with one-hundred independent
replications, to estimate the probability that the network of Problem 4 operates. One
purpose of such a simulation is to check your answer. For more-complicated networks,
however, the simulation might be the only way to analyze the network.
(a) At the top of the sheet, place your name, class, homework number, and problem number.
Leave four or five blank rows.
(b) In Column A place the heading "Trial #", (which is shorter than "Replication #"). Beneath the
heading place the integers 1 through 100. (You can "drag" the integers; you should not
enter the numbers manually.)
(c) In Columns B-G place the headings U1, U2,..., U6. Beneath each of these headings, place
100 random numbers. (Use the "rand" command.)
(d) In Columns H-M place the headings D1, D2,..., D6. Above each of these heading place the
corresponding given probability of the device operating. Beneath each of these six
headings, place 100 binary numbers, 1 if Di occurred (that is, if device i worked) and 0 if
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Homework #4. Due: Wednesday, September 22, 1999.
IE 230
Di did not occur (that is, if device i did not work). (Use the "if" command, just like flipping a
biased coin. The results for Column H should use the random numbers in Column B, I
from C, etc.)
(e) In Column N place the heading "Operates?" Beneath this heading compute the 100 binary
numbers that indicate whether the network operated on each replication. As before, let 1
indicate that the event occurred and 0 that it did not occur. (Use an "if" statement based
on the occurrences in Columns H-M.)
(f) Above the Column N heading, compute the relative frequency that the network operated.
(Use the "average" command.) Label this cell.
(g) Hit F9 enough times to convince yourself that the results match your answer from
Problem 4(d). If the results do not match, check your work. When they do match, write
on your hard copy the relative frequencies obtained from hitting F9 ten times.
(h) Copy your simulation to a new sheet, say Sheet2. Modify your experiment to perform 1000
replications. Write on your hard copy the relative frequencies obtained from hitting F9
ten times. (These ten values should have less variability than the ten values from Part (g).
If not, check your work.)
Quiz Monday, September 20, 1999. The topic is primarily that of homework #3: Sections 3.3-3.5.
You should have memorized the corresponding material from the pages 1-3 of the concise notes.
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Schmeiser