7.4.1 Evaluate the following. Find all the factorials explicitly. ...
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7.4.2 Evaluate the following. Find all the factorials explicitly. ...
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7.4.3 Evaluate the following. Find all the factorials explicitly. ...
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7.4.4 Evaluate the following. Find all the factorials explicitly. ...
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7.4.5 Evaluate the following. Find all the factorials explicitly. ...
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7.4.6 Evaluate the following. Find all the factorials explicitly. ...
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7.4.7 Calculate the given value. b (1; 4, 0.3) (Use the value computed in Exercise 1.) Exercise 1 ...
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7.4.8 Calculate the given value. b (3; 4, 0.3) (Use the value computed in Exercise 2.) Exercise 2 ...
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7.4.9 Calculate the given value. b (2; 5, 0.4) (Use the value computed in Exercise 3.) Exercise 3 ...
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7.4.10 Calculate the given value. b (2; 6, 0.4) (Use the value computed in Exercise 4.) Exercise 4.) ...
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7.4.11 Calculate the given value. b (1; 7, 0.2) (Use the value computed in Exercise 5.) Exercise 5 ...
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7.4.12 Calculate the given value. b (2; 7, 0.6) (Use the value computed in Exercise 6.) Exercise 6 ...
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7.4.13 Find the expectation, variance, and mode for binomial random variables with the following parameters. n =4, p =0.3.
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7.4.14 Find the expectation, variance, and mode for binomial random variables with the following parameters. n =6, p =0.4.
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7.4.15 Find the expectation, variance, and mode for binomial random variables with the following parameters. n =7, p =0.7.
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7.4.16 Find the expectation, variance, and mode for binomial random variables with the following parameters. n =17, p =0.6.
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7.4.17
Use the formula for the binomial probability distribution to find and
graph the probability distribution in the following cases. A binomial
random variable B with p =0.7 and n =2.
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7.4.18
Use the formula for the binomial probability distribution to find and
graph the probability distribution in the following cases. A binomial
random variable B with p =0.4 and n =2.
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7.4.19
Use the formula for the binomial probability distribution to find and
graph the probability distribution in the following cases. A binomial
random variable B with p =0.7 and n =3.
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7.4.20
Use the formula for the binomial probability distribution to find and
graph the probability distribution in the following cases. A binomial
random variable B with p =0.4 and n =3.
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7.4.21 Compute
the mean and the variance from the probability distribution and make
sure that your answers match the formulas in Equations 7.4.2 and 7.4.3. A binomial random variable B with p =0.7 and n =2 (as in Exercise 17).
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7.4.22 Compute
the mean and the variance from the probability distribution and make
sure that your answers match the formulas in Equations 7.4.2 and 7.4.3. A binomial random variable B with p =0.4 and n =2 (as in Exercise 18).
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7.4.23 Compute
the mean and the variance from the probability distribution and make
sure that your answers match the formulas in Equations 7.4.2 and 7.4.3. A binomial random variable B with p =0.7 and n =3 (as in Exercise 19).
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7.4.24 Compute
the mean and the variance from the probability distribution and make
sure that your answers match the formulas in Equations 7.4.2 and 7.4.3. A binomial random variable B with p =0.4 and n =3 (as in Exercise 20).
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7.4.26 Suppose the probability of a success is p.
Find the probability of each of the following events and compare with
the formula for the binomial distribution. List all ways to get two
successes out of four trials, and find the probability of each outcome.
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7.4.26 Suppose the probability of a success is p.
Find the probability of each of the following events and compare with
the formula for the binomial distribution. List all ways to get two
successes out of four trials, and find the probability of each outcome.
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7.4.27 The
binomial distribution depends on two key assumptions: the trials must
each have the same probability of success, and the trials must be
independent. Show that the distribution of outcomes does not match the
binomial distribution in the following cases. Suppose two trials
are independent, but the first has a probability 0.3 of success, and
the second a probability 0.7 of success. Find the probabilities of zero,
one, and two successes, and compare with the binomial distribution with
n =2 and p =0.5 (the average of the two
probabilities). Does the expected number of successes match the binomial
distribution? Does the variance?
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7.4.28 The
binomial distribution depends on two key assumptions: the trials must
each have the same probability of success, and the trials must be
independent. Show that the distribution of outcomes does not match the
binomial distribution in the following cases. Suppose two trials
are independent, but the first has a probability 0.3 of success, and
the second a probability 0.1 of success. Find the probabilities of zero,
one, and two successes, and compare with the binomial distribution with
n =2 and p =0.2. Does the expected number of successes match the binomial distribution? Does the variance?
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7.4.29 The
binomial distribution depends on two key assumptions: the trials must
each have the same probability of success, and the trials must be
independent. Show that the distribution of outcomes does not match the
binomial distribution in the following cases. Suppose the first
trial has probability of success 0.5, and the second is successful with
probability 0.8 if the first is and succeeds with probability 0.2 if the
first fails. Show that the second trial has a probability 0.5 of
success. Find the probabilities of zero, one, and two successes, and
compare with the binomial distribution with n =2 and p =0.5. Does the expected number of successes match the binomial distribution? Does the variance?
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7.4.30 The
binomial distribution depends on two key assumptions: the trials must
each have the same probability of success, and the trials must be
independent. Show that the distribution of outcomes does not match the
binomial distribution in the following cases. Suppose the first
trial has probability of success 0.2, and the second is successful with
probability 0 if the first is and succeeds with probability 0.25 if the
first fails. Show that the second trial has a probability 0.2 of
success. Find the probabilities of zero, one, and two successes, and
compare with the binomial distribution with n =2 and p =0.2.
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7.4.31 The values ...have many beautiful mathematical properties. Here are just a few Show that ... Explain why this must be true.
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7.4.32
The values ...have many beautiful mathematical properties. Here are
just a few Figure out why the following induction should hold, and show
that it does. ...
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7.4.33 The values ...have many beautiful mathematical properties. Here are just a few ...
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7.4.34 The values ...have many beautiful mathematical properties. Here are just a few Explain why the coefficients of the powers of x in the expansion of ...are the binomial coefficients. What is the connection with Exercise 32?
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7.4.35 Suppose
that the probability that a baby is a boy is 0.5 and that a baby is a
girl is also 0.5. Find the probabilities of each of the following
families. Assume that the sexes of babies are independent of each other. Family
C has eight children, seven of whom are girls. Family D also has eight
children, four of whom are girls. Which type of family is more probable?
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7.4.36 Suppose
that the probability that a baby is a boy is 0.5 and that a baby is a
girl is also 0.5. Find the probabilities of each of the following
families. Assume that the sexes of babies are independent of each other. Family
C has eight children: three girls, one boy, and then four more girls.
Family D also has eight children: two girls, one boy, one girl, three
boys, and then a girl. Which type of family is more probable?
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7.4.37 A
group of identical quintuplets named Aaron, Bill, Carl, Dave, and Ed
enjoy confusing the teachers at their school. List the number of
different possibilities in each case, and then count them up. Make sure
your counts match the appropriate value of “n choose k.” Only one goes to school.
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7.4.38 A
group of identical quintuplets named Aaron, Bill, Carl, Dave, and Ed
enjoy confusing the teachers at their school. List the number of
different possibilities in each case, and then count them up. Make sure
your counts match the appropriate value of “n choose k.” Two go to school.
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7.4.39 A
group of identical quintuplets named Aaron, Bill, Carl, Dave, and Ed
enjoy confusing the teachers at their school. List the number of
different possibilities in each case, and then count them up. Make sure
your counts match the appropriate value of “n choose k.” Three go to school.
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7.4.40 A
group of identical quintuplets named Aaron, Bill, Carl, Dave, and Ed
enjoy confusing the teachers at their school. List the number of
different possibilities in each case, and then count them up. Make sure
your counts match the appropriate value of “n choose k.” Four go to school.
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7.4.41
Find the number of ways the following can be ordered. List three of
the possible orderings. The order of finishing by three horses (named
Speedy, Blinky, and Sparky) in a race.
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7.4.42
Find the number of ways the following can be ordered. List three of
the possible orderings. The items in a four-course meal (soup, salad,
main dish, and dessert).
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7.4.43
Find the number of ways the following can be ordered. List three of
the possible orderings. A five-card poker hand (ace, 2, 5, 10, king).
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7.4.44
Find the number of ways the following can be ordered. List three of
the possible orderings. Six occupants of offices along a hall (Al,
Brenda, Carla, Dan, Esther, and Frank).
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7.4.45 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that every one of the patients gets the correct medication?
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7.4.46 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that the first patient gets the right medication?
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7.4.48 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that the first two patients get the right medication?
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7.4.48 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that the first two patients get the right medication?
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7.4.49 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that the third patient
gets the right medication conditional on the first two getting the right
medication? What is the probability that the first three patients get
the right medication?
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7.4.50 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. What is the probability that the second patient
gets the wrong medication conditional on the first getting the wrong
medication? Are the two events independent?
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7.4.51 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. Suppose that four of the patients were prescribed
one medication and the other one was prescribed a different one. What
is the probability that all five get the right medication?
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7.4.52 Each
of five patients has been prescribed a different medication, but the
prescriptions were accidentally shuffled. Compute the following
probabilities. Suppose that three of the patients were
prescribed one medication and the other two were prescribed a different
one. What is the probability that all five get the right medication?
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7.4.53 Suppose a heterozygous plant self-pollinates and produces five offspring with independent genotypes. Find
the probability distribution for the number of heterozygous offspring.
Find the expectation and variance. Sketch the distribution.
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7.4.54 Suppose a heterozygous plant self-pollinates and produces five offspring with independent genotypes. Suppose
that one allele is dominant and produces tall plants. Find and sketch
the probability distribution for the number of tall offspring. Find the
expectation and variance of the number of tall offspring.
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7.4.55
We
have seen that nonindependence of alleles (possibly caused by
differential mortality of genotypes) can lead to deviations from normal
proportions of offspring genotypes. Find the probability that a
surviving offspring from a selfing heterozygote with genotype Aa has
zero, one, or two A alleles. Which cases follow a binomial distribution?
Suppose
that all of the homozygous offspring survive and half of the
heterozygous offspring survive (as in Section 6.2,Exercise 15).
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7.4.56
We
have seen that nonindependence of alleles (possibly caused by
differential mortality of genotypes) can lead to deviations from normal
proportions of offspring genotypes. Find the probability that a
surviving offspring from a selfing heterozygote with genotype Aa has
zero, one, or two A alleles. Which cases follow a binomial distribution?
Suppose that all offspring with genotype AA survive, half of the
offspring with genotype Aa survive, and one fourth of the offspring with
genotype aa survive.
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7.4.57
We
have seen that meiotic drive (where one allele pushes its way into more
than half of the gametes) can lead to deviations from normal
proportions of offspring genotypes. Find the probability that an
offspring from a selfing heterozygote with genotype Aa has zero, one, or
two A alleles. Which cases follow a binomial distribution? Suppose
meiotic drive affects the pollen only and that 80% of the pollen grains
from a heterozygote carry the A allele. Ovules are normal and 50% of
them carry the A allele (as in Section 6.2, Exercise 49).
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7.4.58
We
have seen that meiotic drive (where one allele pushes its way into more
than half of the gametes) can lead to deviations from normal
proportions of offspring genotypes. Find the probability that an
offspring from a selfing heterozygote with genotype Aa has zero, one, or
two A alleles. Which cases follow a binomial distribution? Suppose
meiotic drive affects both pollen and ovules and that 80% of the pollen
grains and ovules from a heterozygote carry the A allele (as in Section
6.2, Exercise 50).
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7.4.59 There is a remarkable approximation for n! called the Stirling approximation (derived by methods related to the method of leading behavior). ...
a. Compare the values for n ranging from 1 to 10.
b. Plot the values of n! and the Stirling approximation for n ranging from 1 to 100 on a semi-log graph. How close are they?
c. Stirling’s formula can be made more accurate by multiplying it by .... How much better is it for n =10? For n =100?
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7.4.60 The following values of n and p all
give an expectation of 10. Use your computer to plot the binomial
distribution in each case. Describe how they are different.
a. n =10, p =1.
b. n =15, p = ... .
c. n =20, p =0.5.
d. n =30, p = ... .
e. n =50, p =0.2.
f. n =100, p =0.1. g. n =1000, p =0.01
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7.4.61 If the probability that a team wins any particular game is 0.6, then the probability that it wins a five-game series is b(3, 5, 0.6) + b(4, 5, 0.6) + b(5, 5, 0.6).
(Think about who would win if teams kept playing even after one team
had won three games.) Figure out how to write this in terms of the
cumulative distribution B, and then how to write it in general for a series of n games with probability q of winning any particular game (assume n is odd). On a single graph, plot the probabilities that a team wins a series of 1 game, 5 games, and 101 games as functions of q. Why is each curve increasing? Explain the shape of the curve with n =1. Explain the shape of the curve with n =101. What is the value of each curve at q =0.5 and why?
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