7.5.1 Suppose that the allele A for height is dominant, meaning
that plants with genotypes AA and Aa are tall, while those with
genotype aa are short. If an Aa plant is crossed with another Aa plant,
3/4 of the offspring should be tall. Assuming that the other conditions
for the binomial distribution are met, find the probabilities of the
following. Exactly three out of four offspring are tall.
Get solution
7.5.2
Suppose that the allele A for height is dominant, meaning that plants
with genotypes AA and Aa are tall, while those with genotype aa are
short. If an Aa plant is crossed with another Aa plant,
3/4 of the offspring should be tall. Assuming that the other conditions
for the binomial distribution are met, find the probabilities of the
following. Exactly six out of eight offspring are tall.
Get solution
7.5.3
Suppose that the allele A for height is dominant, meaning that plants
with genotypes AA and Aa are tall, while those with genotype aa are
short. If an Aa plant is crossed with another Aa plant,
3/4 of the offspring should be tall. Assuming that the other conditions
for the binomial distribution are met, find the probabilities of the
following. One or fewer out of four offspring are tall.
Get solution
7.5.4
Suppose that the allele A for height is dominant, meaning that plants
with genotypes AA and Aa are tall, while those with genotype aa are
short. If an Aa plant is crossed with another Aa plant,
3/4 of the offspring should be tall. Assuming that the other conditions
for the binomial distribution are met, find the probabilities of the
following. Two or fewer out of eight offspring are tall.
Get solution
7.5.5 Compute the coefficient of variation of the binomial distribution in the following cases. p =0.5, n =100.
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7.5.6 Compute the coefficient of variation of the binomial distribution in the following cases. p =0.5, n =25. Why is the coefficient of variation larger than in Exercise 5?
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7.5.7 Compute the coefficient of variation of the binomial distribution in the following cases. p =0.9, n =25.
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7.5.8 Compute the coefficient of variation of the binomial distribution in the following cases. p =0.1, n =25. Why is the coefficient of variation larger than Exercise 7?
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7.5.9
When there are more than two outcomes of a trial, the distribution of
all possibilities is described by the multinomial distribution.
Consider an additive pair of alleles A and a, where an offspring of a
cross between two Aa individuals is tall (with genotypeAA) with
probability 0.25, is intermediate (with genotype Aa) with probability
0.5, and is short (with genotype aa) with probability 0.25. Count up all
possible ways for the following to happen and find the associated
probabilities. Out of three offspring, one is tall, one is
intermediate, and one is short.
Get solution
7.5.10
When there are more than two outcomes of a trial, the distribution of
all possibilities is described by the multinomial distribution.
Consider an additive pair of alleles A and a, where an offspring of a
cross between two Aa individuals is tall (with genotypeAA) with
probability 0.25, is intermediate (with genotype Aa) with probability
0.5, and is short (with genotype aa) with probability 0.25. Count up all
possible ways for the following to happen and find the associated
probabilities. Out of three offspring, one is tall, and two are
intermediate.
Get solution
7.5.11
When there are more than two outcomes of a trial, the distribution of
all possibilities is described by the multinomial distribution.
Consider an additive pair of alleles A and a, where an offspring of a
cross between two Aa individuals is tall (with genotypeAA) with
probability 0.25, is intermediate (with genotype Aa) with probability
0.5, and is short (with genotype aa) with probability 0.25. Count up all
possible ways for the following to happen and find the associated
probabilities. Out of four offspring, two are tall, and two are
intermediate.
Get solution
7.5.13 There
is a formula for the multinomial distribution (Exercises 9–12)
describing probabilities when there are more than two outcomes of a
trial. Suppose there are three possible outcomes of each trial, numbered
1 through 3, with probabilities ..., and .... If there are n trials, the probability that there are exactly ... outcomes of the first type, ... outcomes of the second type, and ...
outcomes of the third type is ... Consider the genetics
described in Exercises 9–12 involving an additive pair of alleles A and
a.
Use the formula for the multinomial distribution to compute the
following probabilities and compare with the earlier result. Out of
three offspring, one is tall, one is intermediate, and one is short (as
in Exercise 9).
Get solution
7.5.14 Out of three offspring, one is tall, and two are intermediate (as in Exercise 10).
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7.5.15 Out of four offspring, two are tall, and two are intermediate (as in Exercise 11).
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7.5.15 Out of four offspring, two are tall, and two are intermediate (as in Exercise 11).
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7.5.16 Out of four offspring, one is tall, two are intermediate, and one is short (as in Exercise 12).
Get solution
7.5.18
Suppose that the alleles A and a for height are additive, meaning
that plants with genotype AA are tall, plants with genotype Aa are
intermediate, and those with genotype aa are short. If an Aa plant is
crossed with another Aa plant,
1/4 of the offspring should be tall, 1/2 should be intermediate, and
1/4 should be short. Assuming that the other conditions for the binomial
distribution are met, find the probabilities of the following if a
cross produces six offspring. Find the expectation and the mode of the
number of intermediate offspring.
Get solution
7.5.19
Suppose that the alleles A and a for height are additive, meaning
that plants with genotype AA are tall, plants with genotype Aa are
intermediate, and those with genotype aa are short. If an Aa plant is
crossed with another Aa plant,
1/4 of the offspring should be tall, 1/2 should be intermediate, and
1/4 should be short. Assuming that the other conditions for the binomial
distribution are met, find the probabilities of the following if a
cross produces six offspring. Find the probability that the number of
tall offspring is less than or equal to the mode.
Get solution
7.5.20
Suppose that the alleles A and a for height are additive, meaning
that plants with genotype AA are tall, plants with genotype Aa are
intermediate, and those with genotype aa are short. If an Aa plant is
crossed with another Aa plant,
1/4 of the offspring should be tall, 1/2 should be intermediate, and
1/4 should be short. Assuming that the other conditions for the binomial
distribution are met, find the probabilities of the following if a
cross produces six offspring. Find the probability that the number of
intermediate offspring is less than or equal to the mode.
Get solution
7.5.20
Suppose that the alleles A and a for height are additive, meaning
that plants with genotype AA are tall, plants with genotype Aa are
intermediate, and those with genotype aa are short. If an Aa plant is
crossed with another Aa plant,
1/4 of the offspring should be tall, 1/2 should be intermediate, and
1/4 should be short. Assuming that the other conditions for the binomial
distribution are met, find the probabilities of the following if a
cross produces six offspring. Find the probability that the number of
intermediate offspring is less than or equal to the mode.
Get solution
7.5.21 Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. The probability that exactly seven out of ten are occupied.
Get solution
7.5.22 Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. The probability that all ten are occupied.
Get solution
7.5.23 Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. Find the mean number of islands occupied.
Get solution
7.5.24 Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. Find the variance of the number of islands occupied.
Get solution
7.5.25 Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. Find the mode of the number of islands occupied.
Get solution
7.5.26
Consider
two sets of ten islands. In the first set, each island has a 0.2 chance
of switching from empty to occupied, and a 0.1 chance of switching from
occupied to empty. The equilibrium fraction occupied is 2/3. In the
second set of ten islands, all are occupied if the weather is good
(probability 2/3) and all empty if the weather is bad (probability 1/3).
Compute the following for both sets of islands. Sketch the probability
distribution for each set of islands. Why does the second set fail to
follow the binomial distribution?
Get solution
7.5.27 Find the probabilities of the following events. Consider
the mutant genes described in Section 6.2, Exercise 27, where a wild
type gene has a 1.0% chance of mutating each time a cell divides and a
mutant gene has a 1.0% chance of reverting to wild type. Suppose that
four genes start out normal. Find the probability that there are two or
more mutants after one division, after two divisions, and after a long
time.
Get solution
7.5.28 Find the probabilities of the following events. Consider
the lemmings described in Section 6.2, Exercise 28, where a lemming has
a probability 0.2 of jumping off the cliff each hour and a probability
0.1 of crawling back up. Suppose that five lemmings start at top. Find
the probability that more than half of the lemmings are at the bottom
after 1 h, after 2 h, and after a long time.
Get solution
7.5.30 Starting
with five molecules, each leaving with probability 0.2 per minute,
compute and graph the probability distribution describing the number
remaining at the following times. 2 min.
Get solution
7.5.30 Starting
with five molecules, each leaving with probability 0.2 per minute,
compute and graph the probability distribution describing the number
remaining at the following times. 2 min.
Get solution
7.5.31 Starting
with five molecules, each leaving with probability 0.2 per minute,
compute and graph the probability distribution describing the number
remaining at the following times. 5 min.
Get solution
7.5.32 Starting
with five molecules, each leaving with probability 0.2 per minute,
compute and graph the probability distribution describing the number
remaining at the following times. 10 min.
Get solution
7.5.33
Starting
with five molecules, each leaving with probability 0.2/min never to
return, find and graph the following probabilities as functions of time.
Exactly one remains. At what time is this probability a maximum?
Get solution
7.5.34
Starting
with five molecules, each leaving with probability 0.2/min never to
return, find and graph the following probabilities as functions of time.
Exactly two remain. At what time is this probability a maximum?
Get solution
7.5.35 Suppose
that ten independent experiments are run, in which five molecules begin
inside a cell and leave with probability 0.2/min and never return. Using
the results in Exercise 29, find the probability that exactly three out
of ten such experiments have exactly four molecules after 1 min.
Get solution
7.5.36 Suppose
that ten independent experiments are run, in which five molecules begin
inside a cell and leave with probability 0.2/min and never return. Using
the results in Exercise 31, find the probability that exactly five out
of ten such experiments have exactly one molecule after 5 min.
Get solution
7.5.37
40 molecules begin inside a cell. Each leaves independently with
probability 0.2/min. Find the expected number remaining inside as a
function of time.
Get solution
7.5.38 40 molecules begin inside a cell. Each leaves independently with probability 0.2/min. Find the mode at t =1, t =2, and t =3. Indicate whether the mode is equal to, greater than, or less than the mean.
Get solution
7.5.39
40 molecules begin inside a cell. Each leaves independently with
probability 0.2/min. Compute the variance. At what time is it a
maximum?
Get solution
7.5.40 40 molecules begin inside a cell. Each leaves independently with probability 0.2/min. Find
the coefficient of variation of the number remaining inside as a
function of time. Is it an increasing or a decreasing function?
Get solution
7.5.41 Example
6.1.5 from Section 6.1, which illustrates stochastic immigration, was
generated by adding two individuals with probability 0.5 and zero
individuals with probability 0.5 for 100 generations. The results in the
figure show final populations of 106 and 96. How can these results be described in terms of the binomial distribution?
Get solution
7.5.42 Example
6.1.5 from Section 6.1, which illustrates stochastic immigration, was
generated by adding two individuals with probability 0.5 and zero
individuals with probability 0.5 for 100 generations. The results in the
figure show final populations of 106 and 96. What is the expected number after 100 generations?
Get solution
7.5.43 Example
6.1.5 from Section 6.1, which illustrates stochastic immigration, was
generated by adding two individuals with probability 0.5 and zero
individuals with probability 0.5 for 100 generations. The results in the
figure show final populations of 106 and 96. What is the variance after 100 generations?
Get solution
7.5.44 Example
6.1.5 from Section 6.1, which illustrates stochastic immigration, was
generated by adding two individuals with probability 0.5 and zero
individuals with probability 0.5 for 100 generations. The results in the
figure show final populations of 106 and 96. Write the probability of exactly 106 in terms of the binomial distribution.
Get solution
7.5.45 Unbeknownst to the experimenter, a cell contains two different types of molecule, one which is inside with probability ... and the other which is inside with probability .... Suppose there are two of each type of molecule. Suppose .... Find and graph the probability distribution for the total number inside. Find the expectation and the variance.
Get solution
7.5.47 Unbeknownst to the experimenter, a cell contains two different types of molecule, one which is inside with probability ... and the other which is inside with probability ....Suppose there are two of each type of molecule. Suppose ....
Find and graph the probability distribution for the total number
inside. Find the expectation and the variance (this can be written as
the sum of two binomial random variables).
Get solution
7.5.47 Unbeknownst to the experimenter, a cell contains two different types of molecule, one which is inside with probability ... and the other which is inside with probability ....Suppose there are two of each type of molecule. Suppose ....
Find and graph the probability distribution for the total number
inside. Find the expectation and the variance (this can be written as
the sum of two binomial random variables).
Get solution
7.5.48 Unbeknownst to the experimenter, a cell contains two different types of molecule, one which is inside with probability ... and the other which is inside with probability .... Suppose there are two of each type of molecule. It turns out that all four molecules are different and that .... Find and graph the probability distribution for the total number inside. Find the expectation and the variance.
Get solution
7.5.49 Simulate
groups of 8 plants, each of which has a 0.25 chance of being tall. How
long does it take until you get half tall plants? If you can automate
the process, try the same with groups of 20 plants.
Get solution
7.5.50 Simulate
groups of 8 plants, each of which has a 0.25 chance of being tall. How
long does it take until you get half tall plants? If you can automate
the process, try the same with groups of 20 plants. The probability ... that a molecule is inside a cell after t time
steps is ... Suppose 100 molecules start out in a cell. Plot
the probabilities that 50 molecules are inside at time t and that 10 are inside at time t as functions of t on
a single graph with reasonable axes. Explain the shape of the curves.
Is it more likely there are exactly 50 or exactly 10 molecules at time
12? What is a more likely number of molecules at this time? Compute the
probability that exactly 10 molecules are inside at time 20 and indicate
this point on your graph. Do the same for the probability that exactly
50 molecules are inside at time 8. Is the area under each curve equal to
1? Why or why not?
Get solution
7.5.51 Simulate
groups of 8 plants, each of which has a 0.25 chance of being tall. How
long does it take until you get half tall plants? If you can automate
the process, try the same with groups of 20 plants. Consider a
situation where two types of molecule leave a cell with different
probabilities during each minute. The first leaves with probability ...and the second leaves with probability .... Suppose we begin with ... of the first type, ...of the second type, and a total of ...If
we did not know the difference, the average probability is ...
Simulate the number of molecules of each type that remain inside after 5
min, and then compare with the number out of n that would remain if they left with the average probability q in each of the following cases. How important is it to distinguish among types of molecule? ...
Get solution
