Download Arnie Pizer Rochester Problem Library Fall 2005

Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability10NormalDist due 06/10/2008 at 02:00am EDT.
1. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 1.pg
Find the following probabilities for the standard normal random
variable z:
(a) P(−0.32 ≤ z ≤ 0.4) =
(b) P(−0.29 ≤ z ≤ 1.93) =
(c) P(z ≤ 1.33) =
(d) P(z > −1.57) =
6.
(1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 8.pg
For a normal distribution, find the percentage of data that are
(a) Within 1.56 standard deviations of the mean.
(b) Between µ − 3σ and µ + 2.5σ
(c) Less than µ − 3.5σ
7. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 5.pg
Suppose the random variable x is best described by a normal
distribution with µ = 30 and σ = 5.4. Find the z-score that corresponds to each of the following x values.
(a) x = 32
z=
(b) x = 37
z=
(c) x = 39
z=
(d) x = 36
z=
(e) x = 19
z=
(f) x = 19
z=
2. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 2.pg
Assume that the readings on the thermometers are normally distributed with a mean of 0◦ and a standard deviation of 1.00◦ C.
A thermometer is randomly selected and tested. Find the probability of each reading in degrees.
(a) Between 0 and 1.64:
(b) Between −1.36 and 0:
(c) Between −0.32 and 0.79:
(b) Less than −1.39:
(c) Greater than 0.72:
3. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 3.pg
Find the value of the standard normal random variable z, called
z0 such that:
(a) P(z ≤ z0 ) = 0.6118
z0 =
(b) P(−z0 ≤ z ≤ z0 ) = 0.4076
z0 =
(c) P(−z0 ≤ z ≤ z0 ) = 0.4442
z0 =
(d) P(z ≥ z0 ) = 0.0616
z0 =
(e) P(−z0 ≤ z ≤ 0) = 0.1001
z0 =
(f) P(−1.95 ≤ z ≤ z0 ) = 0.5125
z0 =
8. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 7.pg
Suppose x is a normally distributed random variable with µ =
10.4 and σ = 1. Find each of the following probabilities:
(a) P(7.1 ≤ x ≤ 13.5) =
(b) P(5.1 ≤ x ≤ 16.1) =
(c) P(11.5 ≤ x ≤ 16.6) =
(d) P(x ≥ 8.5) =
(e) P(x ≤ 14.7) =
9. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 9.pg
The physical fitness of an athlete is often measured by how
much oxygen the athlete takes in (which is recorded in milliliters
per kilogram, ml/kg). The mean maximum oxygen uptake for
elite athletes has been found to be 60 with a standard deviation
of 5.4. Assume that the distribution is approximately normal.
(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least 50 ml/kg?
answer:
(b) What is the probability that an elite athlete has a maximum oxygen uptake of 60 ml/kg or lower?
answer:
(c) Consider someone with a maximum oxygen uptake of
27 ml/kg. Is it likely that this person is an elite athlete? Write
”YES” or ”NO.”
answer:
4. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 4.pg
Assume that the readings on the thermometers are normally idstributed with a mean of 0◦ and a standard deviation of 1.00◦ C.
Find P15 , the 15th percentile.
This is the temperature reading separating the bottom 15 %
from the top 85 %.
5. (1 pt) rochesterLibrary/setProbability10NormalDist/ur pb 10 6.pg
Suppose that the readings on the thermometers are normally distributed with a mean of 0◦ and a standard deviation of 1.00◦C.
If 6% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable,
find the reading that separates the rejected thermometers from
the others.
1
10.
(1
pt)
13.
rochesterLibrary/setProbability10NormalDist-
(1
pt)
rochesterLibrary/setProbability10NormalDist-
/ur pb 10 10.pg
/ur pb 10 13.pg
The combined math and verbal scores for females taking the
SAT-I test are normally distributed with a mean of 998 and
a standard deviation of 202 (based on date from the College
Board). If a college includes a minimum score of 850 among
its requirements, what percentage of females do not satisfy that
requirement?
Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept
track of many lifestyle variables including how many hours of
sleep obtained on a typical day. Researchers found that visually impaired students averaged 9.31 hours of sleep, with a
standard deviation of 2.47 hours. Assume that the number of
hours of sleep for these visually impaired students is normally
distributed.
(a) What is the probability that a visually impaired student
gets less than 6.3 hours of sleep?
answer:
(b) What is the probability that a visually impaired student
gets between 7 and 10.04 hours of sleep?
answer:
(c) Twenty percent of students get less than how many hours
of sleep on a typical day?
answer:
hours
14.
(1 pt) rochesterLibrary/setProbability10NormalDist-
11.
(1
pt)
rochesterLibrary/setProbability10NormalDist-
/ur pb 10 11.pg
The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds
produced from the plant was analyzed for a particular collagen.
The collagen amount was found to be normally distributed with
a mean of 73 and standard deviation of 5.3 grams per mililiter.
(a) What is the probability that the amount of collagen is
greater than 65 grams per mililiter?
answer:
(b) What is the probability that the amount of collagen is less
than 79 grams per mililiter?
answer:
(c) What percentage of compounds formed from the extract
of this plant fall within 3 standard deviations of the mean?
answer:
%
12.
(1
pt)
/ur pb 10 14.pg
Women’s weights are normally distributed with a mean given
by µ = 143 lb and a standard deviation given by σ = 29 lb. Find
the ninth decile, D9 , which separates the bottom 90% from the
top 10%.
rochesterLibrary/setProbability10NormalDist-
15.
/ur pb 10 12.pg
(1
pt)
rochesterLibrary/setProbability10NormalDist-
/ur pb 10 16.pg
IQ scores are normally distributed with a mean of 100 and a
standard deviation of 15. Mensa is an international society that
has one - and only one - qualification for membership: a score
in the top 2on an IQ test.
(a) What IQ score should one have in order to be eligible for
Mensa?
Healty people have body temperatures that are normally distributed with a mean of 98.20◦ F and a standard deviation of
0.62◦ F .
(a) If a healthy person is randomly selected, what is the probability that he or she has a temperature above 98.8◦ F?
answer:
(b) A hospital wants to select a minimum temperature for
requiring further medical tests. What should that temperature
be, if we want only 2.5 % of healty people to exceed it?
answer:
(b) In a typical region of 75,000 people, how many are eligible for Mensa?
c
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Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability11CentralLimitTh due 06/11/2008 at 02:00am EDT.
1.
(1
pt)
• D. When n > 30, the sampling distribution of x will be
approximately a normal distribution.
• E. None of the above
rochesterLibrary/setProbability11CentralLimitTh-
/ur pb 11 1.pg
Assume that women’s weights are normally distributed with a
mean given by µ = 143 lb and a standard deviation given by
σ = 29 lb.
(a) If 1 woman is randomly selected, find the probabity that
her weight is between 113 lb and 177 lb
5.
(1
/ur pb 11 5.pg
(c) If 51 women are randomly selected, find the probability
that they have a mean weight between 113 lb and 177 lb
pt)
rochesterLibrary/setProbability11CentralLimitTh-
Cans of regular Coke are labeled as containing 12 oz.
Statistics students weighted the content of 8 randomly chosen
cans, and found the mean weight to be 12.13.
Assume that cans of Coke are filled so that the actual amounts
are normally distributed with a mean of 12.00 oz and a standard
deviation of 0.11 oz. Find the probability that a sample of 8 cans
will have a mean amount of at least 12.13 oz.
3.
(1
/ur pb 11 3.pg
pt)
(b) If the standard deviation were 3 psi as claimed, but the
mean was 157 psi, what is the probability of obtaining a sample
mean of 156 psi or below?
(c) If the process mean were 157 psi as claimed, but
the standard deviation was 3 psi, what is the probability of obtaining a sample mean of 156 psi or below?
rochesterLibrary/setProbability11CentralLimitTh-
Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of 509 and a standard deviation
of 112.
(a) If 1 man is randomly selected, find the probability that
his score is at least 579.5.
6.
pt)
rochesterLibrary/setProbability11CentralLimitTh-
Suppose that from the past experience a professor knows that
the test score of a student taking his final examination is a random variable with mean 62 and standard deviation 10. How
many students would have to take the examination to ensure,
with probability at least 0.94, that the class average would be
within 3 of 62?
11 randomly selected men were given a review course before taking the SAT test. If their mean score is 579.5, is there a
strong evidence to support the claim that the course is actually
effective?
(Enter YES or NO)
pt)
(1
/ur pb 11 6.pg
(b) If 11 men are randomly selected, find the probability that
their mean score is at least 579.5.
4.
(1
/ur pb 11 4.pg
rochesterLibrary/setProbability11CentralLimitTh-
A soft drink bottler purchases glass bottles from a vendor. The
bottles are required to have an internal pressure of at least 150
pounds per square inch (psi). A prospective bottle vendor claims
that its production process yields bottles with a mean internal
pressure of 157 psi and a standard deviation of 3 psi. The bottler
strikes an agreement with the vendor that permits the bottler to
sample from the production process to verify the claim. the bottler randomly selects 70 bottles from the last 10000 produced,
measures the internal pressure of each, and finds the mean pressure for the sample to be 1 psi below the process mean cited by
the vendor.
(a) Assuming that the vendor is correct in his claim, what is
the probability of obtaining a sample mean this far or farther
below the process mean?
(b) If 3 women are randomly selected, find the probability
that they have a mean weight between 113 lb and 177 lb
2.
(1
/ur pb 11 2.pg
pt)
7.
(1
/ur pb 11 7.pg
pt)
rochesterLibrary/setProbability11CentralLimitTh-
90 numbers are rounded off to the nearest integer and then
summed. If the individual round-off error are uniformly distributed over (−.5, .5) what is the probability that the resultant
sum differs from the exact sum by more than 2?
rochesterLibrary/setProbability11CentralLimitTh-
The Central Limit Theorem says
• A. When n < 30, the original population will be approximately a normal distribution.
• B. When n > 30, the original population will be approximately a normal distribution.
• C. When n < 30, the sampling distribution of x will be
approximately a normal distribution.
8.
(1
/ur pb 11 8.pg
pt)
rochesterLibrary/setProbability11CentralLimitTh-
A die is continuously rolled until the total sum of all rolls exceeds 175. What is the probability that at least 40 rolls are
necessary?
1
c
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Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability12NormApproxBinom due 06/12/2008 at 02:00am EDT.
An airline company is considering a new policy of booking as
many as 382 persons on an airplane that can seat only 330.5.
(Past studies have revealed that only 80% of the booked passengers actually arrive for the flight.) Estimate the probability
that if the company books 382 persons. not enough seats will be
available.
1.
(1 pt) rochesterLibrary/setProbability12NormApproxBinom/ur pb 12 1.pg
Use normal approximation to estimate the probability of getting
exactly 50 girls in 100 births. Assume that boys and girls are
equally likely.
2.
(1 pt) rochesterLibrary/setProbability12NormApproxBinom-
/ur pb 12 2.pg
Use normal approximation to estimate the probability of passing
a true/false test of 70 questions if the minimum passing grade is
90% and all responses are random guesses.
3.
4.
(1 pt) rochesterLibrary/setProbability12NormApproxBinom-
/ur pb 12 4.pg
A multiple-choice test consists of 23 questions with possible
answers of a, b, c, d. Estimate the probability that with random
quessing, the number of correct answers is at least 9.
(1 pt) rochesterLibrary/setProbability12NormApproxBinom-
/ur pb 12 3.pg
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability13UniformDist due 06/13/2008 at 02:00am EDT.
(b)
a=
(c)
a=
(d)
a=
(e)
a=
1. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 1.pg
A manager of an apartment store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 2 to 6 minutes. If it takes the
elevator 15 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than
2.5 minutes after pushing the elevator button on the second floor.
answer :
2. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 2.pg
Suppose the time to process a loan application follows a uniform
distribution over the range 8 to 15 days. What is the probability
that a randomly selected loan application takes longer than 13
days to process?
answer:
3. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 3.pg
Suppose x is a random variable best described by a uniform
probability that ranges from 0 to 5. Compute the following:
(a) the probability density function f (x) =
(b) the mean µ =
(c) the standard deviation σ =
(d) P(µ − σ ≤ x ≤ µ + σ) =
(e) P(x ≥ 4.64) =
P(x < a) = 0.94
P(x ≥ a) = 0.94
P(x > a) = 0.13
P(1.96 ≤ x ≤ a) = 0.33
5. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 5.pg
The weather in Rochester in December is fairly constant.
Records indicate that the low temperature for each day of the
month tend to have a uniform distribution over the interval 15◦
to 35◦ F. A business man arrives on a randomly selected day in
December.
(a) What is the probability that the temperature will be above
26◦ ?
answer:
(b) What is the probability that the temperature will be between 17◦ and 33◦ ?
answer:
(c) What is the expected temperature?
answer:
6. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 8.pg
If a is uniformly distributed over [−4, 18], what is the probability that the roors of the equation
4. (1 pt) rochesterLibrary/setProbability13UniformDist/ur pb 13 4.pg
Suppose a random variable x is best described by a uniform
probability distribution with range 1 to 5. Find the value of a
that makes the following probability statements true.
(a) P(x ≤ a) = 0.76
a=
x2 + ax + a + 3 = 0
are both real?
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability14ExponentialDist due 06/14/2008 at 02:00am EDT.
1.
(1
pt)
Suppose that the life distribution of an item has hazard rate
function λ(t) = 2.7t 2 , t > 0. What is the probability that
(a) the item doesn’t survive to age 1?
(b) the item’s lifetime is between 0.5 and 4?
(c) a 0.5-year-old item will survive to age 4?
rochesterLibrary/setProbability14ExponentialDist-
/ur pb 14 1.pg
Suppose that the time (in hours) required to repair a machine
is an exponentially distributed random variable with parameter
λ = 0.6. What is
(a) the probability that a repair takes less than 8 hours?
3.
(1
pt)
rochesterLibrary/setProbability14ExponentialDist-
/ur pb 14 3.pg
Let X be an exponential random variable with parameter λ = 4,
and let Y be the random variable defined by Y = 10eX . Compute
the probability density function of Y :
fY (t) =
(b) the conditional probability that a repair takes at least 11
hours, given that it takes more than 8 hours?
2.
(1
pt)
rochesterLibrary/setProbability14ExponentialDist-
/ur pb 14 2.pg
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability15OtherContDist due 06/15/2008 at 02:00am EDT.
1.
(1
pt)
The probability density function of X, the lifetime of a certain type of device (measured in months), is given by
0 if x < 13
f (x) =
13
if x > 13
x2
Find the following:
P(X > 33) =
The cumulative distribution function of X:
if x < 13
F(x) =
if x > 13
The probability that at least one out of 3 devices of this type
will function for at least 41 months:
3.
(1 pt) rochesterLibrary/setProbability15OtherContDist-
rochesterLibrary/setProbability15OtherContDist-
/ur pb 15 1.pg
You’ll need to use the formatted text mode in order to do this
problem: click the ”formatted text” button on the bottom of the
page and then click ”submit answers”.
Let X be a random variable with probability density function
c(9x − x2 ) if 0 < x < 9
f (x) =
0
otherwise
Find the value of c:
c=
Find the cumulative distribution function of X:

if x ≤ 0

if 0 < x < 9
F(x) =

if x ≥ 9
2.
(1
/ur pb 15 2.pg
pt)
/ur pb 15 3.pg
The density function of X is given by
a + bx2 if 0 ≤ x ≤ 1
f (x) =
0
otherwise
If the expectation of X is E(X) = −2, find a and b.
a=
b=
rochesterLibrary/setProbability15OtherContDist-
You’ll need to use the formatted text mode in order to do this
problem: click the ”formatted text” button on the bottom of the
page and then click ”submit answers”.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability16JointDist due 06/16/2008 at 02:00am EDT.
1. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 1.pg
The joint probability density function of X and Y is given by
f (x, y) = c(y2 − 400x2 )e−y , −
7. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 7.pg
Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is 2.2. What is the probability that there will be
(a) at least 5 such accidents in the next month?
y
y
≤x≤ , 0<y<∞
20
20
Find c and the expected value of X:
c=
E(X) =
(b) at most 4 such accidents in the next 3 months?
(c) exactly 5 such accidents in the next 4 months?
2. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 2.pg
x and y are uniformly distributed over the interval [0, 1]. Find the
probability that |x − y|, the distance between x and y, is less than
0.8.
8. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 8.pg
Sam’s bowling scores are approximately normally distributed
with mean 170 and standard deviation 23, while John’s scores
are normally distributed with mean 135 and standard deviation
16. If Sam and John each bowl one game, then assuming that
their scores are independent random variables, approximate the
probability that the total of their scores is above 290.
3. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 3.pg
A man and a woman agree to meet at a cafe about noon. If
the man arrives at a time uniformly distributed between 11 : 35
and 12 : 10 and if the woman independently arrives at a time
uniformly distributed between 11 : 40 and 12 : 25, what is the
probability that the first to arrive waits no longer than 5 minutes?
9. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 9.pg
The joint probability mass function of X and Y is given by
4. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 4.pg
Two points are selected randomly on a line of length 12 so as to
be on opposite sides of the midpoint of the line. In other words,
the two points X and Y are independent random variables such
that X is uniformly distriuted over [0, 6) and Y is uniformly distributed over (6, 12]. Find the probability that the distance between the two points is greater than 3.
answer:
5.
Let
p(1, 1) = 0.25
p(2, 1) = 0.1
p(3, 1) = 0.1
f (x) =
cx5 y8
0
p(1, 3) = 0.1
p(2, 3) = 0.1
p(3, 3) = 0
(a) Compute the conditional mass function of Y given X = 2:
P(Y = 1|X = 2) =
P(Y = 2|X = 2) =
P(Y = 3|X = 2) =
(b) Are X and Y independent? (enter YES or NO)
(c) Compute the following probabilities:
P(X +Y > 3) =
P(XY = 2) =
P( YX > 1) =
(1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 5.pg
p(1, 2) = 0.1
p(2, 2) = 0.2
p(3, 2) = 0.05
if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
otherwise
Find the following:
(a) c such that f (x, y) is a probability density function:
c=
(b) Expected values of X and Y :
E(X) =
E(Y ) =
(c) Are X and Y independent? (enter YES or NO)
10. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 9a.pg
The joint probability mass function of X and Y is given by
p(1, 1) = 0.4
p(2, 1) = 0.1
p(3, 1) = 0.1
6. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 6.pg
Let A, B, and C be independent random variables, uniformly
distributed over [0, 10], [0, 7], and [0, 6] respectively. What is
the probability that both roots of the equation Ax2 + Bx +C = 0
are real?
p(1, 2) = 0.05
p(2, 2) = 0.05
p(3, 2) = 0.05
Compute the following probabilities:
P(X +Y > 4) =
P(XY = 4) =
P( YX > 2) =
1
p(1, 3) = 0.1
p(2, 3) = 0.1
p(3, 3) = 0.05
P(Y = 2|X = 1) =
P(Y = 3|X = 1) =
(b) Are X and Y independent? (enter YES or NO)
11. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 9b.pg
The joint probability mass function of X and Y is given by
p(1, 1) = 0.05
p(2, 1) = 0.05
p(3, 1) = 0.05
p(1, 2) = 0.05
p(2, 2) = 0.05
p(3, 2) = 0.1
12. (1 pt) rochesterLibrary/setProbability16JointDist/ur pb 16 10.pg
Two points along a straight stick of length 46cm are randomly
selected. The stick is then broken at those two points. Find the
probability that all of the resulting pieces have lenght at least
2cm.
p(1, 3) = 0.1
p(2, 3) = 0.1
p(3, 3) = 0.45
(a) Compute the conditional mass function of Y given X = 1:
P(Y = 1|X = 1) =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability17Expectation due 06/17/2008 at 02:00am EDT.
1. (1 pt) rochesterLibrary/setProbability17Expectation/ur pb 17 1.pg
A fair die is rolled 5 times. What is the expected sum of the 5
rolls?
3. (1 pt) rochesterLibrary/setProbability17Expectation/ur pb 17 3.pg
Consider n = 25 independent flips of a fair coin. Say that
a changeover occurs whenever an outcome differs from the
one preceding it. For example, if n = 6 and the outcome is
T H T T H T, then there is a total of 4 changeorvers. Find
the expected number of changeovers for n = 25.
2. (1 pt) rochesterLibrary/setProbability17Expectation/ur pb 17 2.pg
29 people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among
those present. That person then either sits at the table of a friend
or at an unoccupied table is none
of those present is a friend.
29
Assuming that each of the 2 pairs of people are, independently, friends with probability 0.7, find the expected number of
occupied tables.
4. (1 pt) rochesterLibrary/setProbability17Expectation/ur pb 17 4.pg
If E[X] = 0 and Var(X) = 4, then
E[(4 + 3X)2 ] =
and
.
Var(4 + 5X)2 =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability1Combinations due 06/01/2008 at 02:00am EDT.
8. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 8.pg
A computer retail store has 9 personal computers in stock. A
buyer wants to purchase 4 of them. Unknown to either the retail
store or the buyer, 4 of the computers in stock have defective
hard drives. Assume that the computers are selected at random.
(a) In how many different ways can the 4 computers be chosen?
answer:
(b) What is the probability that exactly one of the computers
will be defective?
answer:
(c) What is the probability that at least one of the computers
selected is defective?
answer:
1. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 1.pg
Find the value of the permutation:
P(7, 5) =
2. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 2.pg
Find the value of the combination:
C(11, 5) =
3. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 3.pg
How many 3-digit numbers can be formed using the digits 1, 3,
5, 7, and 9? Repeated digits are allowed.
4. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 4.pg
How many different 11-letter words (real or imaginary) can be
formed from the letters in the word PROBABILITY?
9. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 9.pg
In how many ways can 5 novels, 3 mathematics books, and 1
biology book be arranged on a bookshelf if
(a) the books can be arranged in any order?
answer:
(b) the mathematics books must be together and the novels
must be together?
answer:
(c) the mathematics books must be together but the other
books can be arranged in any order?
answer:
5. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 5.pg
Determine the size of the sample space that corresponds to the
experiment of tossing a coin the following number of times:
(a) 2 times
answer:
(b) 6 times
answer:
(c) n times
answer:
10.
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur pb 1 10.pg
From a group of 6 women and 8 men a committee consisting
of 3 men and 4 women is to be formed. How many different
committees are possible if
(a) 2 of the men refuse to serve together?
answer:
(a) 2 of the women refuse to serve together?
answer:
(a) 1 man and 1 woman refuse to serve together?
answer:
6. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 6.pg
An experiment consists of choosing objects without regards to
order. Determine the size of the sample space when you choose
the following:
(a) 3 objects from 15
answer :
(b) 6 objects from 19
answer :
(c) 2 objects from 28
answer :
11.
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur dis 9 1.pg
(a) A particular brand of shirt comes in 14 colors, has a male
version and a female version, and comes in 3 sizes for each sex.
How many different types of this shirt are made?
7. (1 pt) rochesterLibrary/setProbability1Combinations/ur pb 1 7.pg
Suppose you are managing 20 employees, and you need to form
three teams to work on different projects. Assume that all employees will work on a team, and that each employee has the
same qualifications/skills so that everyone has the same probability of getting choosen. In how many different ways can
the teams be chosen so taht the number of employees on each
project are as follows:
9, 4, 7
answer :
(b) How many bit strings of length 9 are there?
(c) How many bit strings of length 9 or less are there?
(Count the empty string of length zero also.)
(d) How many strings of 6 lower case English letters are there
that have the letter x in them somewhere? Here strings may use
1
the same letter more than once. (Hint: It might be easier to first
count the strings that don’t have an x in them.)
16.
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur dis 9 6.pg
12.
(1
pt)
A bowl contains 6 red balls and 6 blue balls. A woman selects
balls at random without looking at them.
(a) How many balls must she select (minimum) to be sure of
having at least three blue balls?
(b) How many balls must she select (minimum) to be sure of
having at least three balls of the same color?
rochesterLibrary/setProbability1Combinations-
/ur dis 9 2.pg
Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive,
have the following properties:
(a) have distinct digits.
17.
(b) are not divisible by either 5 or 7.
(d) are divisible by 5 and by 7.
(1
pt)
pt)
rochesterLibrary/setProbability1Combinations-
This question concerns bit strings of length six. These bit strings
can be divided up into four types depending on their initial
and terminal bit. Thus the types are: 0XXXX0, 0XXXX1,
1XXXX0, 1XXXX1.
How many bit strings of length six must you select before you
are sure to have at least 5 that are of the same type? (Assume
that when you select bit strings you always select different ones
from ones you have already selected.)
(c) are divisible by 5.
13.
(1
/ur dis 9 7.pg
rochesterLibrary/setProbability1Combinations-
/ur dis 9 3.pg
How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can
start with zeros.)
(a) end with an even digit? (Can repeat digits.)
18.
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur dis 9 8.pg
/ur dis 9 4.pg
Find the value of each of the following quantities:
C(11, 10) =
C(9, 8) =
C(9, 5) =
C(5, 3) =
C(6, 5) =
C(5, 2) =
How many strings of five uppercase English letters are there
(a) that start with an X, if no letter can be repeated?
/ur dis 9 9.pg
(b) that start and end with the letters BO (in that order), if
letters can be repeated?
There are 4 different candidates for governor of a state. In how
many different orders can the names of the candidates be printed
on a ballot?
(b) begin with an odd digit? (can repeat digits.)
14.
(1
pt)
rochesterLibrary/setProbability1Combinations-
19.
(c) that start with an X, if letters can be repeated?
20.
(1
pt)
(1
pt)
pt)
rochesterLibrary/setProbability1Combinations-
rochesterLibrary/setProbability1Combinations-
/ur dis 9 10.pg
(d) that start or end with the letters BO (in the order), if letters
can be repeated? (inclusive or)
15.
(1
How many bit strings of length 8 have:
(a) Exactly three 0s?
(b) The same number of 0s as 1s?
(d) At least three 1s?
rochesterLibrary/setProbability1Combinations-
21.
/ur dis 9 5.pg
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur dis 9 11.pg
Solve the following two “ union ” type questions:
(a) How many bit strings of length 7 either begin with 1 0s or
end with 1 1s? (inclusive or)
16 players for a softball team show up for a game:
(a) How many ways are there to choose 10 players to take the
field?
(b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these
two subjects. How many students are in the class if there are
33 computer science majors (including joint majors), 22 math
majors (including joint majors) and 5 joint majors?
(b) How many ways are there to assign the 10 positions by
selecting players from the 13 people who show up?
(c) Of the 16 people who show up, 6 are women. How many
ways are there to choose 10 players to take the field if at least
one of these players must be women?
2
22.
(1
pt)
23.
rochesterLibrary/setProbability1Combinations-
(1
pt)
rochesterLibrary/setProbability1Combinations-
/ur dis 9 13.pg
/ur dis 9 12.pg
How many ways are there to select 9 countries in the United
Nations to serve on a council if 2 is selected from a block of 56,
2 are selected from a block of 68 and 5 are selected from the
remaining 65 countries?
Suppose that a department contains 12 men and 15 women.
How many ways are there to form a committee with 6 members
if it must have strictly more women than men?
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability2BinomialTh due 06/02/2008 at 02:00am EDT.
1. (1 pt) rochesterLibrary/setProbability2BinomialTh/ur
pb 2 1.pg
Evaluate the binomial coefficient: 14
7
4. (1 pt) rochesterLibrary/setProbability2BinomialTh/ur dis 9 14.pg
Find the coefficient of x8 in (1 + x)10 .
2. (1 pt) rochesterLibrary/setProbability2BinomialTh/ur pb 2 2.pg
Expand the expression using the Binomial Theorem:
x5 +
x4 +
x3 +
x2 +
x+
(3x − 3)5 =
3.
5. (1 pt) rochesterLibrary/setProbability2BinomialTh/ur dis 9 15.pg
What is the coefficient of x6 y13 in the expansion of (−3x −
1y)19 ?
(1 pt) rochesterLibrary/setProbability2BinomialTh/ur pb 2 3.pg
Find the coefficient of
x−5
5
in the expansion of 5x −
x
2
8
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability3Events due 06/03/2008 at 02:00am EDT.
1.
(a) P(C) =
(b) P(A ∩ B) =
(c) P(A ∪C ∪ D) =
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 1.pg
The sample space for an experiment contains five sample
points. The probabilities of the sample points are:
P(1) = P(2) = 0.25
P(3) = P(4) = 0.1
P(5) = 0.3
Find the probability of each of the following events:
A : { Either 4 or 5 occurs }
B : { Either 4, 2, or 3 occurs }
C : {1 does not occur }
P(B) =
P(C) =
P(A) =
2.
5.
A couple decided to have 3 children.
(a) What is the probability that they will have at least two
girls?
(b) What is the probability that all the children will be of the
same gender?
6.
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 8.pg
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 2.pg
Consider two people being randomly selected. (For simplicity, ignore leap years.)
(a) What is the probability that two people born in September have a birthday in the first half of the month?
answer:
(b) What is the probability that two people have a birthday
on the same day of the same month?
answer:
Two fair dice are tossed, and the up face on each die is
recorded. Find the probability of observing each of the following events:
A : { The sum of the numbers is odd }
B : { The sum of the numbers is equal to 6 }
C : { A 1 does not appear on either die }
P(A) =
P(B) =
P(C) =
3.
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 7.pg
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 3.pg
7.
Consider the experiment composed of one roll of a fair die
followed by one toss of a fair coin. Determine the probability of
each of the following events.
A : { An odd number appears on the die. }
B : { An even number appears on the die; a T appears on the
coin. }
C : { An H appears on the coin. }
P(A) =
P(B) =
P(C) =
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 10.pg
In a study by the Department of Transportation, there were
a total of 86 drivers that were pulled over for speeding. Out of
those 86 drivers, 34 were men who were ticketed, 15 were men
who were not ticketed, 6 were women who were ticketed, and
31 were women who were not ticketed. Suppose one person was
chosen at random.
(a) What is the probability that the selected person is a
woman or someone who was ticketed?
answer:
(b) What is the probability that the selected person is a
woman or someone who was not ticketed?
answer:
4. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 5.pg
In the game Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal lenght, numbered 00, 0, 1, 2, ... ,
35, 36. The number on the arc on which the ball stops is the
outcome of one play of the game. The numbers are also colored
as follows:
1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are
red,
2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35
are black,
0, 00 are green
Define the following events:
A : { Outcome is an even number (0 and 00 are considered neither odd nor even) }
B : { Outcome is a red number }
C : { Outcome is a green number }
D : { Outcome is a low number (1-18) }
Find the following probablilities:
8. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 12.pg
An elementary school is offering 3 language classes: one in
Spanish, one in French, and one in German. These classes are
open to any of the 78 students in the school. There are 33 in
the Spanish class, 27 in the French class, and 15 in the German
class. There are 10 students that in both Spanish and French, 5
are in both Spanish and German, and 6 are in both French and
German. In addition, there are 2 students taking all 3 classes.
If one student is chosen randomly, what is the probability that
he or she is taking at least two language classes?
If two students are chosen randomly, what is the probability
that neither of them is taking a language class?
1
9. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 13.pg
A group of kids containing 17 boys and 12 girls is lined up in
random order - that is, each of the 29! permutations is assumed
to be equally likely. What is the probability that the person in
the 8-th position is a boy?
10. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 14.pg
An instructor gives his class a set of 13 problems with the information that the next quiz will consist of a random selection of 5
of them. If a student has figured out how to do 9 of the problems,
what is the probability the he or she will answer correctly
(a) all 5 problems?
(b) at least 4 problems?
11. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 15.pg
How many people have to be in a room in order that the probability that at least two of them celebrate their birthday on the
same day is at least 0.2? (Ignore leap years, and assume that all
outcomes are equally likely.)
12.
Use the Venn diagram and the probabilities of the sample
points to find:
(a) P(Bc ) =
(b) P(Ac ∩ B) =
(c) P(A ∪ B) =
(d) P(Ac ) =
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 4.pg
15. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 6a.pg
A sample space contains 7 sample points and events A and B as
seen in the Venn diagram.
Let P(1) = P(2) = P(3) = P(7) = 0.1
P(4) = P(5) = 0.05
and P(6) = 0.5.
A fair coin is tossed three times and the events A, B, and C
are defined as follows:
A : { At least one head is observed }
B : { At least two heads are observed }
C : { The number of heads observed is odd }
Find the following probabilities by summing the probabilities of the appropriate sample points:
(a) P(B) =
(b) P(Ac ∩ B) =
(c) P(A ∪ B ∪C) =
13.
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 4a.pg
A fair coin is tossed three times and the events A, B, and C
are defined as follows:
A : { At least one head is observed }
B : { At least two heads are observed }
C : { The number of heads observed is odd }
Find the following probabilities by summing the probabilities of the appropriate sample points:
(a) P(B) =
(b) P(A ∩C) =
(c) P(A ∪ B ∪C) =
Use the Venn diagram and the probabilities of the sample
points to find:
(a) P(B) =
(b) P(A ∩ B) =
(c) P(A ∪ B) =
(d) P(A ∪ A) =
14. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 6.pg
A sample space contains 7 sample points and events A and B as
seen in the Venn diagram.
Let P(1) = P(2) = P(3) = P(7) = 0.05
P(4) = P(5) = 0.1
and P(6) = 0.6.
2
16.
(1 pt) rochesterLibrary/setProbability3Events/ur pb 3 9.pg
(c) What is the probability that the sum on the two dice comes
out to be 11?
The number 35 is written as a sum of three natural numbers
(d) What is the probability that the numbers on the two dice
are equal?
35 = a + b + c
(the triple (a, b, c) is ordered; e.g., the decompositions 35 =
1 + 1 + 33 and 35 = 1 + 33 + 1 are different.
Also, assume that all the decompositions have equal probability.)
What is the probability that there exists a triangle with sides
a, b, and c?
17. (1 pt) rochesterLibrary/setProbability3Events/ur pb 3 11.pg
A quick quiz consists of 4 multiple choice problems, each of
which has 6 answers, only one of which is correct. If you make
random guesses on all 4 problems,
(a) What is the probability that all 4 of your answers are
incorrect?
answer:
(b) What is the probability that all 4 of your answers are
correct?
answer:
18. (1 pt) rochesterLibrary/setProbability3Events/ur dis 9 16.pg
Two six-sided dice are rolled (one red one and one green one).
Some possibilities are (Red=1,Green=5) or (Red=2,Green=2)
etc.
(a) How many total possibilities are there?
A card is selected at random from a standard 52-card deck.
(a) What is the probability that it is an ace?
(b) What is the probability that it is a heart?
(c) What is the probability that is an ace or a heart?
20. (1 pt) rochesterLibrary/setProbability3Events/ur dis 9 18.pg
A five-card poker hand is dealt at random from a standard 52card deck.
Note the total number of possible hands is C(52,5)=2,598,960.
Find the probabilities of the following scenarios:
(a) What is the probability that the hand contains exactly one
ace?
α
, where α =
Answer= C(52,5)
(b) What is the probability that the hand is a flush? (That is
all the cards are of the same suit: hearts, clubs, spades or diamonds.)
β
Answer= C(52,5)
, where β =
(c) What is the probability that the hand is a straight flush?
γ
Answer= C(52,5)
, where γ =
For the rest of the questions, we will assume that the dice are
fair and that all of the possibilities in (a) are equally likely.
(b) What is the probability that the sum on the two dice comes
out to be 10? (Remember the answer will be a ratio, the denominator of which will be your answer in (a).)
What is the probability that a positive integer m in the range
1 ≤ m ≤ 100, which is selected randomly, is divisible by 3?
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability4Conditional due 06/04/2008 at 02:00am EDT.
3.
1.
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 2a.pg
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 1.pg
If P(A) = 0.2, P(B) = 0.9, and P(A ∩ B) = 0.15, then
(a) P(A|B) =
and
(b) P(B|A) =
2.
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 2.pg
A sample space contains six sample points and events A, B, and
C as shown in the Venn diagram. The probablities of the sample
points are P(1) = 0.1, P(2) = 0.15, P(3) = 0.15, P(4) = 0.3,
P(5) = 0.05, P(6) = 0.25.
Use the Venn diagram and the probabilities of the sample points
to find:
(a) P(B) =
(b) P(B|A) =
(c) P(C|A) =
4.
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 3.pg
A box contains one yellow, two red, and three green balls. Two
balls are randomly chosen without replacement. Define the following events:
A : { One of the balls is yellow }
B : { At least one ball is red }
C : { Both balls are green }
D : { Both balls are of the same color }
Find the following conditional probabilities:
(a) P(A|Bc ) =
(b) P(Dc |B) =
(c) P(C|D) =
A sample space contains six sample points and events A, B, and
C as shown in the Venn diagram. The probablities of the sample
points are P(1) = 0.15, P(2) = 0.5, P(3) = 0.1, P(4) = 0.05,
P(5) = 0.05, P(6) = 0.15.
Use the Venn diagram and the probabilities of the sample points
to find:
(a) P(Bc ) =
(b) P(A|C) =
(c) P(Bc |Cc ) =
5.
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 3a.pg
A box contains one yellow, two red, and three green balls. Two
balls are randomly chosen without replacement. Define the following events:
A : { One of the balls is yellow }
B : { At least one ball is red }
C : { Both balls are green }
D : { Both balls are of the same color }
1
Find the following conditional probabilities:
(a) P(A|B) =
(b) P(B|D) =
(c) P(C|D) =
6.
9.
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 6.pg
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 4.pg
”Channel One” is an educational television network for which
participating secondary schools are equipped with TV sets in
every classroom. It has been found that 70
% of secondary schools subscribe to Channel One, where of
these subscribers 10
% never use Channel One while 15
%claim to use it more than 5 times per week.
Find the probability that a randomly selected seconday
school subscribes to Channel One and uses it more than 5 times
per week.
answer:
7.
A sample space contains six sample points and events A, B, and
C as shown in the Venn diagram. The probablities of the sam1
1
4
3
ple points are P(1) = 12
, P(2) = 12
, P(3) = 12
, P(4) = 12
,
2
1
P(5) = 12 , P(6) = 12 .
Are events A and C mutually exclusive?
• A. Yes
• B. No
Use the Venn diagram and the probabilities of the sample
points to find:
P(A) =
P(B) =
P(A ∩ B) =
Are events A and B independent?
• A. Yes
• B. No
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 5.pg
Two fair dice, one blue and one red, are tossed, and the up face
on each die is recorded. Define the following events:
E : { The sum of the numbers is even }
F : { The numbers are equal }
Find the following probabilities:
(a) P(E) =
(b) P(F) =
(c) P(E ∩ F) =
(d) P(E|F) =
(e) P(F|E) =
Are events E and F independent?
• A. yes
• B. no
8.
10. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 7.pg
Scoring a hole-in-one is the greatest shot a golfer can make.
Once 2 professional golfers each made holes-in-one on the 5th
hole at the same golf course at the same tournament. It has been
found that the estimated probability of making a hole-in-one is
1
2846 for male professionals. Suppose that a sample of 2 professional male golfers is randomly selected.
(a) What is the probability that none of these golfers make a
hole-in-one on the 10th hole at the same tournament?
answer:
(b) What is the probability that at least one of these golfers
makes a hole-in-one on the 10th hole at the same tournament?
answer:
11. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 8.pg
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 5a.pg
Two fair dice, one blue and one red, are tossed, and the up face
on each die is recorded. Define the following events:
E : { The sum of the numbers is even }
F : { The numbers are equal }
Find the following probabilities:
(a) P(E) =
(b) P(F) =
(c) P(E ∩ F) =
Are events E and F independent?
• A. no
• B. yes
For two events A and B, P(A) = 0.2 and P(B) = 0.3.
(a) If A and B are independent, then
P(A ∩ B) =
P(A ∪ B) =
2
P(A|B) =
(b) If A and B are dependent and P(A|B) = 0.15, then
P(A ∩ B) =
P(B|A) =
12.
What is the conditional probability that a randomly selected
family owns a dog given that it owns a cat?
16. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 13.pg
Urn A has 6 white and 16 red balls. Urn B has 5 white and 12
red balls. We flip a fair coin. If the oucome is heads, then a ball
from urn A is selected, whereas if the oucome is tails, then a
ball from urn B is selected. Suppose that a red ball is selected.
What is the probability that the coin landed heads?
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 9.pg
If P(A) = 0.4, P(B) = 0.5, and P(A ∪ B) = 0.9, then
P(A ∩ B) =
.
(a) Are events A and B independent? (enter YES or NO)
(b) Are A and B mutually exclusive? (enter YES or NO)
13.
17. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 14/ur pb 4 14.pg
(1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 10.pg
The probability of the closing of the ith relay in the circuits
shown is given by pi . Let p1 = 0.4, p2 = 0.6, p3 = 0.3, p4 = 0.8,
p5 = 0.2. If all relays function independently. what is the probability that a current flows between A and B for the respective
circuits?
The number 34 is written as a sum of three natural numbers
34 = a + b + c
(the triple (a, b, c) is ordered; e.g., the decompositions 34 =
9 + 10 + 15 and 34 = 10 + 15 + 9 are different.
Also, assume that all the decompositions have equal probability.)
Given that there exists a triangle with sides a, b, and c, what
is the probability that this triangle is isosceles?
14. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 11.pg
What is the probability that at least one of a pair of fair dice
lands of 5, given that the sum of the dice is 7?
15. (1 pt) rochesterLibrary/setProbability4Conditional/ur pb 4 12.pg
In a certain community, 16% of the families own a dog, and 25%
of the families that own a dog also own a cat. It is also know
that 33% of all the families own a cat.
What is the probability that a randomly selected family owns
both a dog and a cat?
(a) P =
(b) P =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability5RandomSample due 06/05/2008 at 02:00am EDT.
1.
(1
pt)
2.
rochesterLibrary/setProbability5RandomSample-
(1
pt)
rochesterLibrary/setProbability5RandomSample-
/ur pb 5 1.pg
/ur pb 5 2.pg
(a) Count the number of ways to arrange a sample of 2 elements
from a population of 10 elements.
answer:
(b) If random sampling is to be employed, the probability
that any particular sample will be selected is
A financial firm is performing an assessment test and relies on a
random sampling of their accounts. Suppose this firm has 6559
customer accounts numbered from 0001 to 6559.
One account is to be chosen at random. What is the probability that the selected account number is 1058?
answer:
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability7RandomVariables due 06/07/2008 at 02:00am EDT.
1.
5.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 8.pg
(1 pt) rochesterLibrary/setProbability7RandomVariables-
/ur pb 7 1.pg
Determine whether the following are valid probability distributions or not. Type ”VALID” if it is valid, or type ”INVALID” if
it is not a valid probability distributions.
(a)
Let x represent the difference between the number of heads and
the number of tails when a coin is tossed 42 times. Then
P(x = 2) =
6.
x
2
3
6
8
P(x) 0.6 0.2 0.1 0.1
(1 pt) rochesterLibrary/setProbability7RandomVariables-
/ur pb 7 9.pg
Four buses carrying 156 high school students arrive to Montreal.
The buses carry, respectively, 31, 41, 34, and 50 students. One
of the studetns is randomly selected. Let X denote the number
of students that were on the bus carrying this randomly selected
student. One of the 4 bus drivers is also randomly selected.
Let Y denote the number of students on his bus. Compute the
expectations of X and Y :
E(X) =
E(Y ) =
answer:
(b)
1
P(x) = 5x
, where x = 1, 2, 3, ...
answer:
(c)
x
2 3
6
8
P(x) 0 0.1 −0.2 1.1
answer:
2.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 2.pg
7.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 9a.pg
The mean and standard deviation of a random variable x are −6
and 2 respectively. Find the mean and standard deviation of the
given random variables:
(1) y = x + 8
µ=
σ=
(2) v = 6x
µ=
σ=
(3) w = 6x + 8
µ=
σ=
Four buses carrying 153 high school students arrive to Montreal.
The buses carry, respectively, 37, 49, 27, and 40 students. One
of the studetns is randomly selected. Let X denote the number
of students that were on the bus carrying this randomly selected
student. One of the 4 bus drivers is also randomly selected.
Let Y denote the number of students on his bus. Compute the
expectations and variances of X and Y :
E(X) =
Var(X) =
E(Y ) =
Var(Y ) =
8.
3.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 3.pg
(1 pt) rochesterLibrary/setProbability7RandomVariables-
/ur pb 7 4.pg
Two fair dice are rolled 5 times. Let the random variable x
represent the number of times that the sum 3 occurs. The table
below describes the probability distribution. Find the value of
the missing probability.
x
6
7
8
9
10
P(x) 0.2 0.2 0.1 0.1 0.4
Given the discrete probability distribution above, determine
the following:
(a) P(x = 6) =
(b) P(x ≥ 9 or x < 7) =
(c) P(x ≤ 9) =
4.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 7.pg
x
P(x)
0
0.75141884282545
1
2 0.0260006520008806
3 0.00152945011769886
4 4.49838269911429e-05
5 5.29221494013446e-07
Three dice are rolled. Let the random variable x represent the
sum of the 3 dice. By assuming that each of the 63 possible
outcomes is equally likely, find the probability that x equals 11.
P(x = 11) =
Would it be unusual to roll a pair of dice 5 times and get no
3s?
(enter YES or NO)
1
10.
9.
(1 pt) rochesterLibrary/setProbability7RandomVariables/ur pb 7 5.pg
(1 pt) rochesterLibrary/setProbability7RandomVariables-
/ur pb 7 6.pg
Prizes and the chances of winning in a sweepstakes are given in
the table below.
Prize
Chances
$20,000,000
1 chance in 400,000,000
$250,000
1 chance in 100,000,000
$50,000
1 chance in 10,000,000
$15,000
1 chance in 2,000,000
$500,000
1 chance in 400,000
A watch valued at $70
1 chance in 4,000
A rock concert producer has scheduled an outdoor concert. The
producer estimates the attendance will depend on the weather
according to the following table.
Weather Attendance Probability
wet, cold
5000
0.1
wet, warm
20000
0.2
dry, cold
15000
0.1
dry, warm
45000
0.6
(a) Find the expected value (in dollars) of the amount won
by one entry.
(a) What is the expected attendace?
answer:
(b) If tickets cost $ 25 each, the band will cost $ 150000,
plus $ 50000 for administration. What is the expected profit?
answer:
(b) Find the expected value (in dollars) if the cost of entering
this sweepstakes is the cost of a postage stamp (34 cents)
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability8BinomialDist due 06/08/2008 at 02:00am EDT.
5. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 5.pg
If x is a binomial random variable, compute P(x) for each of the
following cases:
(a) P(x ≤ 4), n = 8, p = 0.6
P(x) =
(b) P(x > 6), n = 9, p = 0.7
P(x) =
(c) P(x < 2), n = 3, p = 0.9
P(x) =
(d) P(x ≥ 4), n = 5, p = 0.6
P(x) =
1. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 1.pg
If x is a binomial random variable, compute p(x) for each of the
following cases:
(a) n = 6, x = 6, p = 0.1
p(x) =
(b) n = 5, x = 3, p = 0.4
p(x) =
(c) n = 3, x = 1, p = 0.2
p(x) =
(d) n = 5, x = 4, p = 0.3
p(x) =
6.
2. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 2.pg
The rates of on-time flights for commercial jets are continuously
tracked by the U.S. Department of Transportation. Recently,
Southwest Air had the best reate with 80 % of its flights arriving
on time. A test is conducted by randomly selecting 13 Southwest flights and observing whether they arrive on time.
(a) Find the probability that exactly 11 flights arrive on time.
(1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 6.pg
The Census Bureau reports that 82% of Americans over the
age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1370 who were over
the age of 25, and 1130 of them were high school graduates.
(a) Find the mean and standard deviation for the number of
high school graduates in groups of 1370 Americans over the age
of 25.
Mean =
Standard deviation =
(b) Is that county result of 1130 unusually high, or low, or
neither?
(Enter HIGH or LOW or NEITHER)
(b) Would it be unusual for Southwest to have 4 flights arrive
late?
(Enter YES or NO)
3. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 3.pg
If x is a binomial random variable, compute the mean, the standard deviation, and the variance for each of the following cases:
(a) n = 5, p = 0.2
µ=
σ2 =
σ=
(b) n = 6, p = 0.5
µ=
σ2 =
σ=
(c) n = 4, p = 0.6
µ=
σ2 =
σ=
(d) n = 3, p = 0.5
µ=
σ2 =
σ=
4. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 4.pg
A quiz cosists of 20 multiple-choice questions, each with 4 possible answers. For someone who makes random guesses for all
of the answers, find the probability of passing if the minimum
passing grade is 70 %.
7. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 7.pg
To determine whether or not they have a certain desease, 100
people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group
the people in groups of 10. The blood samples of the 10 people in each group will be pooled and analized together. If the
test is negative. one test will suffice for the 10 people (we are
assuming that the pooled test will be positive if and only if at
least one person in the pool has the desease); whereas, if the test
is positive each of the 10 people will also be individually tested
and, in all, 11 tests will be made on this group. Assume the
probability that a person has the desease is 0.08 for all people,
independently of each other, and compute the expected number
of tests necessary for each group.
answer:
8. (1 pt) rochesterLibrary/setProbability8BinomialDist/ur pb 8 8.pg
A man claims to have extrasensory perception. As a test, a fair
coin is flipped 28 times, and the man is asked to predict the outcome in advance. He gets 21 out of 28 correct. What is the
probability that he would have done at least this well if he had
no ESP?
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment Probability9PoissonDist due 06/09/2008 at 02:00am EDT.
1. (1 pt) rochesterLibrary/setProbability9PoissonDist/ur pb 9 1.pg
Given that x is a random variable having a Poisson distribution,
compute the following:
(a) P(x = 8) when µ = 4
P(x) =
(b) P(x ≤ 9)when µ = 4
P(x) =
(c) P(x > 4) when µ = 5
P(x) =
(d) P(x < 9) when µ = 3.5
P(x) =
3. (1 pt) rochesterLibrary/setProbability9PoissonDist/ur pb 9 3.pg
The mean number of patients admitted per day to the emergency
room of a small hospital is 3. If, on any given day, there are
only 1 beds available for new patients, what is the probability
that the hospital will not have enough beds to accommodate its
newly admitted patients?
answer:
4. (1 pt) rochesterLibrary/setProbability9PoissonDist/ur pb 9 4.pg
A certain typing agency employs two typists. The average number of errors per article is 3.6 when typed by the first typist and
1.3 when typed by the second. If your article is equally likely to
be typed by either typist, find the probability that it will have no
errors.
2. (1 pt) rochesterLibrary/setProbability9PoissonDist/ur pb 9 2.pg
A statistics professor finds that when she schedules an office
hour for student help, an average of 2.2 students arrive. Find the
probability that in a randomly selected office hour, the number
of student arrivals is 7.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1