6.4.1 For
the given sample spaces, find a set of mutually exclusive and
collective exhaustive events with the given number of elements. S ={0, 1, 2, 3, 4}. Find a set of two mutually exclusive and collective exhaustive events.
Get solution
6.4.2 For
the given sample spaces, find a set of mutually exclusive and
collective exhaustive events with the given number of elements. S ={0, 1, 2, 3, 4}. Find a set of three mutually exclusive and collective exhaustive events.
Get solution
6.4.3 For
the given sample spaces, find a set of mutually exclusive and
collective exhaustive events with the given number of elements. S ={1, 2, 3, 4, . . .}, the set of all positive integers. Find a set of two mutually exclusive and collective exhaustive events.
Get solution
6.4.4 For
the given sample spaces, find a set of mutually exclusive and
collective exhaustive events with the given number of elements. S ={1, 2, 3, 4, . . .}, the set of all positive integers. Find a set of three mutually exclusive and collective exhaustive events.
Get solution
6.4.5 In each of the following cases, find Pr(A ∩ B), Pr(A | B), and Pr(B |A). As in Section 6.3, Exercise 5, the sample space is S={0, 1, 2, 3, 4}, Pr({0})=0.2, Pr({1})=0.3, Pr({2})=0.4, Pr({3})=0.1, Pr({4})=0.0, A={0, 1, 2}, and B={0, 2, 4}.
Get solution
6.4.6 In each of the following cases, find Pr(A ∩ B), Pr(A | B), and Pr(B |A). As in Section 6.3, Exercise 5, the sample space is S={0, 1, 2, 3, 4}, Pr({0})=0.2, Pr({1})=0.3, Pr({2})= 0.4, Pr({3})=0.1, and Pr({4})=0.0. Suppose now that A={1, 2, 3} and B={2, 3, 4}.
Get solution
6.4.7 In each of the following cases, find Pr(A ∩ B), Pr(A | B), and Pr(B |A). As in Section 6.3, Exercise 6, the sample space is S={0, 1, 2, 3, 4}, Pr({0})=0.1, Pr({1})=0.3, Pr({2})=0.4, Pr({3})=0.1, Pr({4})=0.1, A={0, 2}, and B={3, 4}.
Get solution
6.4.8 In each of the following cases, find Pr(A ∩ B), Pr(A | B), and Pr(B |A). As in Section 6.3, Exercise 6, the sample space is S={0, 1, 2, 3, 4}, Pr({0})=0.1, Pr({1})=0.3, Pr({2})=0.4, Pr({3})=0.1, and Pr({4})=0.1. Suppose now that A={1, 2} and B={1, 2, 3, 4}.
Get solution
6.4.9 Somebody
invents a three-sided die that gives scores of 1, 2, or 3, each with
probability 1/3. Two such die are rolled. Use the law of total
probability to find the following probabilities, and then find them
directly by counting. The probability that the total on the two
die is 4 or more. (To use the law of total probability, find the
probability that the score is 4 or more if the first die gives a 1, if
the first die gives a 2, and if the first die gives a 3.)
Get solution
6.4.10 Somebody
invents a three-sided die that gives scores of 1, 2, or 3, each with
probability 1/3. Two such die are rolled. Use the law of total
probability to find the following probabilities, and then find them
directly by counting. The probability that the total on the two die is 5 or more.
Get solution
6.4.11 Somebody
invents a three-sided die that gives scores of 1, 2, or 3, each with
probability 1/3. Two such die are rolled. Use the law of total
probability to find the following probabilities, and then find them
directly by counting. The probability that the total on the two die is odd.
Get solution
6.4.12 Somebody
invents a three-sided die that gives scores of 1, 2, or 3, each with
probability 1/3. Two such die are rolled. Use the law of total
probability to find the following probabilities, and then find them
directly by counting. The probability that the second roll was larger than the first.
Get solution
6.4.13 Consider
again the three-sided die that gives a score of 1 with probability 1/3,
a score of 2 with probability 1/3, and a score of 3 with probability
1/3. Two such die are rolled. Use Bayes’ theorem to find the following
conditional probabilities. Check your result by direct counting. Find
the probability that the first roll is a 3 if the total of the two
rolls is greater than or equal to 4 (based on Exercise 9).
Get solution
6.4.14 Consider
again the three-sided die that gives a score of 1 with probability 1/3,
a score of 2 with probability 1/3, and a score of 3 with probability
1/3. Two such die are rolled. Use Bayes’ theorem to find the following
conditional probabilities. Check your result by direct counting. Find
the probability that the first roll is a 3 if the total of the two
rolls is greater than or equal to 5 (based on Exercise 10).
Get solution
6.4.15
Consider
again the three-sided die that gives a score of 1 with probability 1/3,
a score of 2 with probability 1/3, and a score of 3 with probability
1/3. Two such die are rolled. Use Bayes’ theorem to find the following
conditional probabilities. Check your result by direct counting. Find
the probability that the first roll is a 3 if the total of the two rolls
is odd (based on Exercise 11).
Get solution
6.4.16
Consider
again the three-sided die that gives a score of 1 with probability 1/3,
a score of 2 with probability 1/3, and a score of 3 with probability
1/3. Two such die are rolled. Use Bayes’ theorem to find the following
conditional probabilities. Check your result by direct counting. Find
the probability that the first roll is a 1 if the second roll is greater
than the first (based on Exercise 12).
Get solution
6.4.17
Four balls are placed in a jar, two red, one blue, and one yellow.
Two are removed at random. You are told that the first ball removed was
red. What is the probability that the second is red?
Get solution
6.4.18
Four balls are placed in a jar, two red, one blue, and one yellow.
Two are removed at random. You are told that at least one of the two
removed is red. What is the probability that both are red?
Get solution
6.4.19 Four balls are placed in a jar, two red, one blue, and one yellow. Two are removed at random. As
in Exercise 17, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that the second is
red?
Get solution
6.4.20 Four balls are placed in a jar, two red, one blue, and one yellow. Two are removed at random. As
in Exercise 18, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that both are red?
Get solution
6.4.21
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
17. Reference Exercise 17. You are told that the first ball removed
was red. What is the probability that the second is red?
Get solution
6.4.22
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
18. Reference Exercise 18. You are told that at least one of the two
removed is red. What is the probability that both are red?
Get solution
6.4.23
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
19. Reference Exercise 19. As
in Exercise 17, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that the second is
red?
Get solution
6.4.24
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
20. Reference Exercise 20. As
in Exercise 18, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that both are red?
Get solution
6.4.25
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
23. Reference Exercise 23. The situation in Exercise 19. Reference
Exercise 19. As
in Exercise 17, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that the second is
red?
Get solution
6.4.26
Give a set of three mutually exclusive and collectively exhaustive
sets for each of the following sample spaces. The situation in Exercise
24. Reference Exercise 24. The situation in Exercise 20. Reference
Exercise 20. As
in Exercise 18, but the first ball is replaced (but remembered) before
the second ball is drawn. What is the probability that both are red?
Get solution
6.4.27 An
ecologist is looking for the effects of eagle predation on the behavior
of jackrabbits. In each of the following cases,
a. Draw a Venn diagram
to illustrate the situation.
b. Find the probability that she
saw a jackrabbit conditional on her seeing an eagle. How might you
interpret this result? Compare with the overall probability of seeing a
jackrabbit.
c. Find the probability that she saw an eagle conditional on her seeing a jackrabbit. How might you interpret this result? She
sees an eagle with probability 0.2 during an hour of observation, a
jackrabbit with probability 0.5, and both with probability 0.05.
Get solution
6.4.28 An
ecologist is looking for the effects of eagle predation on the behavior
of jackrabbits. In each of the following cases,
a. Draw a Venn diagram
to illustrate the situation.
b. Find the probability that she
saw a jackrabbit conditional on her seeing an eagle. How might you
interpret this result? Compare with the overall probability of seeing a
jackrabbit.
c. Find the probability that she saw an eagle conditional on her seeing a jackrabbit. How might you interpret this result? She
sees an eagle with probability 0.2 during an hour of observation, a
jackrabbit with probability 0.5, and both with probability 0.15.
Get solution
6.4.29
A
lab is attempting to stain many cells. Young cells stain properly 90%
of the time and old cells stain properly 70% of the time. If 30% of the
cells are young, what is the probability that a cell stains properly?
Get solution
6.4.30
A
lab is attempting to stain many cells. Young cells stain properly 90%
of the time and old cells stain properly 70% of the time. If 70% of the
cells are young, what is the probability that a cell stains properly?
Get solution
6.4.31 Further
study of the cell-staining problem (Exercises 29 and 30) reveals that
new cells stain properly with probability 0.95, 1-day-old cells stain
properly with probability 0.9, 2-day-old cells stain properly with
probability 0.8, and 3-day-old cells stain properly with probability
0.5. Suppose Pr(cell is 0 day old) = 0.4 Pr(cell is 1 day old) = 0.3 Pr(cell is 2 days old) = 0.2 Pr(cell is 3 days old) = 0.1. Find the probability that a cell stains properly.
Get solution
6.4.32 Further
study of the cell-staining problem (Exercises 29 and 30) reveals that
new cells stain properly with probability 0.95, 1-day-old cells stain
properly with probability 0.9, 2-day-old cells stain properly with
probability 0.8, and 3-day-old cells stain properly with probability
0.5. Suppose Pr(cell is 0 day old) = 0.4 Pr(cell is 1 day old) = 0.3 Pr(cell is 2 days old) = 0.2 Pr(cell is 3 days old) = 0.1. The
lab finds a way to eliminate the oldest cells (more than 3 days old)
from its stock. What is the probability of proper staining? Write this
as a conditional probability.
Get solution
6.4.33 Use Bayes’ theorem to compute the following. Say whether the stain is a good indicator of the age of the cell. For
the cells in Exercise 29, what is the probability that a cell that
stains properly is young? How does this compare with the unconditional
probability of 0.3?
Get solution
6.4.34 Use Bayes’ theorem to compute the following. Say whether the stain is a good indicator of the age of the cell. For
the cells in Exercise 30, what is the probability that a cell that
stains properly is young? How does this compare with the unconditional
probability of 0.7?
Get solution
6.4.35 Use Bayes’ theorem to compute the following. Say whether the stain is a good indicator of the age of the cell. For
the cells in Exercise 31, what is the probability that a cell that
stains properly is less than 1 day old? How does this compare with the
unconditional probability of 0.4?
Get solution
6.4.36 Use Bayes’ theorem to compute the following. Say whether the stain is a good indicator of the age of the cell. For
the cells in Exercise 32, what is the probability that a cell that
stains properly is less than 1 day old? How does this compare with the
unconditional probability?
Get solution
6.4.37 Consider
a disease with an imperfect test. Let D denote the event of an
individual having the disease, N the event of not having the disease,
and P the event of a positive result on the test. In each of the
following cases, find Pr(D | P). Pr(D)=0.2, Pr(N)=0.8, Pr(P |D)=1.00, and Pr(P |N)= 0.05. Compare with the results in the text, when the disease was much less common.
Get solution
6.4.38 Consider
a disease with an imperfect test. Let D denote the event of an
individual having the disease, N the event of not having the disease,
and P the event of a positive result on the test. In each of the
following cases, find Pr(D | P). Pr(D)=0.8, Pr(P |D)=1.00, and Pr(P |N)=0.1. Compare with the results in the text and Exercise 37.
Get solution
6.4.39 In
the following cases, the test does not catch every sick person. Let D
denote the event of an individual having the disease, N the event of not
having the disease, and P the event of a positive result on the test.
Find Pr(D | P) and Pr(D|Pc) (the probability that a person who did not test positive has the disease). Pr(D)=0.2, Pr(N)=0.8, Pr(P |D)=0.95, and Pr(P |N)= 0.05. Compare your results with Exercise 37.
Get solution
6.4.40 In
the following cases, the test does not catch every sick person. Let D
denote the event of an individual having the disease, N the event of not
having the disease, and P the event of a positive result on the test.
Find Pr(D | P) and Pr(D|Pc) (the probability that a person who did not test positive has the disease). Pr(D)=0.8, Pr(P |D)=0.95, and Pr(P |N)=0.1. Compare your results with Exercise 38.
Get solution
6.4.41
Consider a dominant gene where plants with genotype BB or Bb are
tall, while plants with genotype bb are short. Find the probability that
a tall plant has genotype Bb when it results from the following
crosses. A plant with genotype Bb is crossed with the offspring from a
cross between a BB plant and a Bb plant.
Get solution
6.4.42
Consider a dominant gene where plants with genotype BB or Bb are
tall, while plants with genotype bb are short. Find the probability that
a tall plant has genotype Bb when it results from the following
crosses. A plant with genotype Bb is crossed with the offspring from a
cross between a Bb plant and a Bb plant.
Get solution
6.4.43
Consider a dominant gene where plants with genotype BB or Bb are
tall, while plants with genotype bb are short. Find the probability that
a tall plant has genotype Bb when it results from the following
crosses. Two offspring from the cross between a BB plant and a Bb plant
are crossed with each other.
Get solution
6.4.44
Consider a dominant gene where plants with genotype BB or Bb are
tall, while plants with genotype bb are short. Find the probability that
a tall plant has genotype Bb when it results from the following
crosses. Two tall offspring from the cross between a Bb plant and a Bb
plant are crossed with each other.
Get solution
6.4.45 A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. If the car is really behind door 1, what happens if she switches?
Get solution
6.4.46 A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. If the car is really behind door 2, what happens if she switches?
Get solution
6.4.47 A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. Should the contestant switch her guess to door 2?
Get solution
6.4.48 A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. If she uses the right strategy, what is her probability of getting the new car?
Get solution
6.4.49 A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. It is later revealed that Monty
does not always show what is behind one of the other doors, but does so
only when the contestant guessed right in the first place (the
so-called “Machiavellian Monty”). How often would a contestant who used
the strategy in Exercise 47 get the new car?
Get solution
6.4.50
A
popular probability problem refers to a once popular game show called
“Let’s Make a Deal.” In this game, the host (named Monty Hall) hands out
large prizes to contestants for no reason at all. In one situation,
Monty would show the contestant three doors, named door 1, door 2, and
door 3. One would hide a new car, one $500 worth of false eyelashes, and
the other a goat (deemed worthless by the purveyors of the show). The
contestant picks door 1. But instead of showing her the prize, Monty
opens door 3 to reveal the goat. What is the right strategy to use for
dealing with the Machiavellian Monty? How well would the contestant do?
Get solution
6.4.51 Use the command ... (Section 6.1, Exercise 37) that returns 1 with probability p and 0 with probability 1 − p to simulate the rare disease example.
a. Simulate 100 people who have the disease with probability 0.05 and count up the number with the disease. b.
For each of the remaining people, assume that the probability of a
false positive is 0.1. Simulate them and count up the number of
positives.
c. What fraction of positive tests identify people who are sick? How does this compare with the mathematical expectation?
d. Try the same experiment where the probability that each person has the disease is 0.4.
Get solution
6.4.52 Suppose
cells fall into three categories: those that are dead, those that are
alive but do not stain properly, and those that are alive and do stain
properly. Let ... denotes the number of cells that are dead, ... the number that are alive but do not stain properly, and ... those
that are alive and do stain properly. Each day, there are two possible
transitions. • Cells that are alive die with probability 0.9. • Cells
that stain properly cease to stain properly with probability 0.8.
a. Start with .... Use your computer to simulate the numbers in the next generation.
b. Follow these cells until there are no more cells that stain properly. How long did it take?
c. At each time, what is the fraction of cells that stain properly? What is the fraction of living cells
that stain properly? Estimate the probability that a cell stains
properly and the probability it stains properly conditional on
beingalive.
Get solution