HOMEWORK ASSIGNMENTS Smith Section ST 371

HOMEWORK and READING ASSIGNMENTS
for ST371-002 Smith Section SPRING 2004

HW # 4 Due Thursday Feb 26, 2004


1. Assume that for single lanches of a space shuttle,
that there is a constant probability = 0.10
of the failure of an O-ring.
For the next 10 future lanches, let X be the NUMBER of those lanches having an O-ring failure.
a) What type of probability model is appropriate,
say the NAME of the probability mass function and give VALUES
for the parameters.
b) Find Pr[X = 2 ]
c) Find Pr[ X >= 1 ]  where >= means greater than or equal
d) Find E[X]  namely the mean or expected value of X
e) Find Var[X] the variance of X
f) Find the standard deviation of X
2. Transmission line interuptions in a telecommunications network
occur at an AVERAGE RATE of 0.5 PER DAY
a) Use the Poisson distribution as a model for the random variable
X = the number of interruptions in the next 5-day work week.
and FIND Pr [ X = 0], Pr[ X > 2], Pr[ X < 2 ]
b) Now consider the random variable Y, where
Y = the number of WEEKS in the next 4 workweeks in which there
are NO interruptions
i) what is a reasonable probability model (which prob mass function
and parameters)
ii) Find Pr[Y=2], Pr[Y=0]
3. Consider the following experiment, 10 subjects are all asked to taste
a soft drink from five unmarked cups. One cup contains Dr Pepper and the other
4 contain a store brand version of Dr Pepper, a fake or clone Dr Pepper,
such as the 4 cups might be Dr Extreme (Harris Teeter), Dr Perfect (Bi-Lo)
Dr. Perky (Food Lion)and Dr Whatever (Wellspring)
The subjects are then asked to write down which cup
contains the real Dr Pepper. If the subjects are purely guessing,
the chances of that person saying the right cup is 0.20 = 1/5
a) What is the appropriate probabilty mass function and parameter values
for this experiment IF the subjects are purely guessing, i.e. can't taste the difference
Let X = the number of subjects picking the true Dr Pepper cup
b) make a probability table for the experiment under the hypothesis that they are purely guessing

X
________________________________________________________
p(x)
c) Find the E[X] and Var[X]
d) If you ran the experiment on 10 of your friends and 7 or more
them correctly identified the Dr Pepper cup, is this strong evidence
that they can taste the difference. Explain your reasoning.
difference between the name brand and fake ones?
4. A committee of size 5 is to be selected at random from 4 computer scientists
and 6 electrical engineers.
a)Find the probability mass function for X=the NUMBER of computer scientists
on the committee and make a table for it
b)Find the E[X] and Var[X]
5. A manufactor of automobile tires reports that among a shipment of 5000 sent
to a local distributor in Lizard Lick, NC that 500 are slightly blemished
If you purchase 10 of these tires at random (it is a cold rainy day)
what is the probability that exactly 3 will be slightly blemished?
6. It is estimated that 5000 of the 10,000 voting residents of a small
town are AGAINST a new sales tax. If 15 eligible voters are selected at random
and asked their opinion, what is the probability that AT MOST 7 favor
the new sales tax? (Hint: use the binomial approximation)
7. Suppose that the probability is 0.7 that any given person will believe the
tale or rumor about the transgressions of a famous actress.
what is the probability that :
a) the SIXTH person to hear this rumor is the FOURTH one to believe it?
b) the THIRD person to hear this rumor is the FIRST one to believe it?
8. In a manufactoring process in which glass items are being produced
defects or bubbles occur occasionally which make that piece of glass unsuitable
for marketing. It is known that ON AVERAGE 2 in every 1000 of these items
produced is defective in this way.
What is the probablity that a random sample of 8000 will yield
FEWER THAN 7 items having defects?
9. A certain area of the eastern US is , on average, hit by 5 hurricanes
in a year. Find the probability that in a given year this area will be hit by
a) FEWER than 4 hurricanes
b) anywhere from 6 to 8 hurricanes
10. Hermione is giving out free samples of Kiwi Broccolli chewing gum
to students watching a sporting match at Hogwarts school.
Assume there is a 75 % chance any given student will refuse to take the
sample of gum .
a) What is the probabilty she gives away her second sample
to the fifth student she meets?
b) what are the mean and variance of the NUMBER of students who refuse
to take the gum before she successfully gives out her second sample
extra credit: (10 points)
For reasons not entirely clear, Klutz airlines flies a daily shuttle service 
from Kalamazoo Michigan to Kankakee, Illinois.

They offer two round trip flights per day. One flight is on a

TWO  engine prop plane , while the other flight is on a FOUR engine plane.


SUPPOSE that each engine on each plane will fail

 independently with the same probability p. 

Also assume that each plane will arrive safely at its destination ONLY if AT 
LEAST HALF of its engines remain in working order.


Further assume that you would like to continue living, for what range of values 
of p would you prefer to fly in the two engine plane?