Solutions Modeling Dynamics of Life 3ed Adler - Chapter 6.5

6.5.1 Check whether the following events are independent by checking three equations: Pr(A) = Pr(A | B)     A is independent of B Pr(B) = Pr(B |A)    B is independent of A Pr(A ∩ B) = Pr(A)Pr(B).      the multiplication rule Do you ever find a case where only one or two of these equations is satisfied? As in Section 6.3, Exercise 5 and Section 6.4, 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}.
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6.5.2 Check whether the following events are independent by checking three equations: Pr(A) = Pr(A | B)     A is independent of B Pr(B) = Pr(B |A)    B is independent of A Pr(A ∩ B) = Pr(A)Pr(B).      the multiplication rule Do you ever find a case where only one or two of these equations is satisfied? As in Section 6.3, Exercise 5 and Section 6.4, Exercise 6, 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={1, 2, 3}, and B={2, 3, 4}.
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6.5.3 Check whether the following events are independent by checking three equations: Pr(A) = Pr(A | B)     A is independent of B Pr(B) = Pr(B |A)    B is independent of A Pr(A ∩ B) = Pr(A)Pr(B).      the multiplication rule Do you ever find a case where only one or two of these equations is satisfied? The sample space is S={1, 2, 3, 4}, Pr({1})=0.48, Pr({2})= 0.12, Pr({3})=0.32, Pr({4})=0.08, A={3, 4}, and B={1, 3}.
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6.5.4 Check whether the following events are independent by checking three equations: Pr(A) = Pr(A | B)     A is independent of B Pr(B) = Pr(B |A)    B is independent of A Pr(A ∩ B) = Pr(A)Pr(B).      the multiplication rule Do you ever find a case where only one or two of these equations is satisfied? The sample space is S={1, 2, 3, 4}, Pr({1})=0.4, Pr({2})= 0.4, Pr({3})=0.1, Pr({4})=0.1, A={1, 2}, and B={1, 3}.
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6.5.5 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling a 1 followed by a 3.
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6.5.6 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling three 1’s in a row.
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6.5.7 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling two odd values in a row.
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6.5.8 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling an odd value followed by an even value.
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6.5.9 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling six 3’s in a row.
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6.5.10 Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3 (Section 6.4, Exercises 9–12). Suppose that the results of rolls are independent. Use the multiplication rule to find the following probabilities. The probability of rolling exactly the sequence 1, 2, 3, 3, 2, 1.
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6.5.11 In each of the following problems, the sample space is S ={1, 2, 3, 4}. From the probabilities of the given events A and B, and the assumption that A and B are independent, find Pr({1}), Pr({2}), Pr({3}), and Pr({4}). A={1, 2}, B={1, 3}, Pr(A)=0.4, Pr(B)=0.6.
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6.5.12 In each of the following problems, the sample space is S ={1, 2, 3, 4}. From the probabilities of the given events A and B, and the assumption that A and B are independent, find Pr({1}), Pr({2}), Pr({3}), and Pr({4}). A={1, 4}, B={1, 3}, Pr(A)=0.8, Pr(B)=0.3.
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6.5.13 Show that the multiplication rule (Theorem 6.3) does not work in the following cases. For two events A and B that are disjoint, as long as Pr(A)>0 and Pr(B)>0.
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6.5.14 Show that the multiplication rule (Theorem 6.3) does not work in the following cases. For two events A and B where A is a subset of B, as long as Pr(A)>0 and Pr(B)<1.
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6.5.15 Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a discretetime dynamical system for the probability, and find the long-term probability. The mutants described in Section 6.2, Exercise 27, where a 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.
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6.5.16 Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a discretetime dynamical system for the probability, and find the long-term probability. 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.
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6.5.17 Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a discretetime dynamical system for the probability, and find the long-term probability. The molecules described in Section 6.2, Exercise 29, where a molecule has a probability 0.05 of binding and a probability of 0.02 of unbinding each second.
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6.5.19 The formula for the probability of the union of two events, Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A ∩ B), (from Section 6.3, Exercises 13–16) is simpler when events are independent. Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A)Pr(B). Test the formula on the following independent events. Using the probabilities found in Exercise 11, where A={1, 2}, B={1, 3}, Pr(A)=0.4, Pr(B)=0.6.
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6.5.19 The formula for the probability of the union of two events, Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A ∩ B), (from Section 6.3, Exercises 13–16) is simpler when events are independent. Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A)Pr(B). Test the formula on the following independent events. Using the probabilities found in Exercise 11, where A={1, 2}, B={1, 3}, Pr(A)=0.4, Pr(B)=0.6.
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6.5.20 The formula for the probability of the union of two events, Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A ∩ B), (from Section 6.3, Exercises 13–16) is simpler when events are independent. Pr(A ∪ B)=Pr(A) + Pr(B) − Pr(A)Pr(B). Test the formula on the following independent events. Using the probabilities found in Exercise 12, where A={1, 4}, B={1, 3}, Pr(A)=0.8, Pr(B)=0.3.
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6.5.22 An ecologist is looking for the effects of eagle predation on the behavior of jackrabbit (as in Section 6.4, Exercises 27 and 28). Assuming that the jackrabbits and eagles behave independently,
a. Find the probability that the ecologist sees both a jackrabbit and an eagle during a particular hour of observation.
b. Draw a Venn diagram to illustrate the situation. c. 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.
d. 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 0.4 during an hour of observation and a jackrabbit with probability 0.8.
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6.5.22 An ecologist is looking for the effects of eagle predation on the behavior of jackrabbit (as in Section 6.4, Exercises 27 and 28). Assuming that the jackrabbits and eagles behave independently,
a. Find the probability that the ecologist sees both a jackrabbit and an eagle during a particular hour of observation.
b. Draw a Venn diagram to illustrate the situation. c. 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.
d. 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 0.4 during an hour of observation and a jackrabbit with probability 0.8.
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6.5.23 Someone comes up with a cut-rate “test” for a disease. This test gives a positive result with probability 0.5 whether or not the patient has the disease. In each of the following cases, find the probability of having the disease conditional on a positive test in two ways.
a. Work it out directly as on page 539.
b. Use independence. 1% of people have the disease.
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6.5.24 Someone comes up with a cut-rate “test” for a disease. This test gives a positive result with probability 0.5 whether or not the patient has the disease. In each of the following cases, find the probability of having the disease conditional on a positive test in two ways.
a. Work it out directly as on page 539.
b. Use independence. 10% of people have the disease.
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6.5.25 Consider again the molecules in Section 6.2, Exercises 1–4. Suppose that we wish to consider two molecules instead of one molecule, both starting inside the cell. Find the following probabilities. What is the probability that both of the molecules remain inside after 1 second (using parameters from Section 6.2, Exercise 1)?Reference Exercise 1For the given probability that a molecule leaves a cell, write the discrete-time dynamical system for the probability that it remains inside (assuming it can never return) and find the solution. Compute the probability that the molecule remains inside after 10 seconds, and the time before it will have left with probability 0.9. The probability it leaves is 0.3 each second.
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6.5.26 Consider again the molecules in Section 6.2, Exercises 1–4. Suppose that we wish to consider two molecules instead of one molecule, both starting inside the cell. Find the following probabilities. What is the probability that both of the molecules have moved outside after 1 second (using parameters from Section 6.2, Exercise 2)?Reference Section 6.2, Exercise 2 For the given probability that a molecule leaves a cell, write the discrete-time dynamical system for the probability that it remains inside (assuming it can never return) and find the solution. Compute the probability that the molecule remains inside after 10 seconds, and the time before it will have left with probability 0.9. The probability it leaves is 0.03 each second.
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6.5.27 Consider again the molecules in Section 6.2, Exercises 1–4. Suppose that we wish to consider two molecules instead of one molecule, both starting inside the cell. Find the following probabilities. What is the probability that both of the molecules remain inside after 2 seconds (using parameters from Section 6.2, Exercise 3)?Reference Exercise 3 The following probabilities describe molecules that can hop into and out of a cell. For each, find a discrete-time dynamical system for the probability that the molecule is inside. Find the probability that a molecule that begins inside is inside at t =2, and the probability that a molecule that begins outside is inside at t =2. Compute the equilibrium, and use it to estimate how many out of 100 molecules would be inside after a long time. The probability it leaves is 0.3 each second, and the probability it returns is 0.2 each second.
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6.5.28 Consider again the molecules in Section 6.2, Exercises 1–4. Suppose that we wish to consider two molecules instead of one molecule, both starting inside the cell. Find the following probabilities. What is the probability that both of the molecules have moved outside after 2 seconds (using parameters from Section 6.2, Exercise 4)?Reference Section 6.2, Exercise 4 The following probabilities describe molecules that can hop into and out of a cell. For each, find a discrete-time dynamical system for the probability that the molecule is inside. Find the probability that a molecule that begins inside is inside at t =2, and the probability that a molecule that begins outside is inside at t =2. Compute the equilibrium, and use it to estimate how many out of 100 molecules would be inside after a long time. The probability it leaves is 0.03 each second, and the probability it returns is 0.1 each second.
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6.5.29 One force that can alter the ratio of heterozygotes produced by a selfing heterozygote is meiotic drive (Section 6.2, Exercises 49–52), where one allele, say A, pushes its way into more than half of the gametes. Another possibility is that the alleles in surviving offspring are not independent. Compare the fraction of heterozygotes produced in the following cases. Compare a case of meiotic drive where 60% of both pollen and ovules carry the A allele independently, with a case of nonindependent assortment where an offspring gets an A allele from the pollen with probability 0.6 when the ovule provides an A and gets an A allele from the pollen with probability 0.4 when the ovule provides an
a. The ovule provides A with probability 0.5.
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6.5.30 One force that can alter the ratio of heterozygotes produced by a selfing heterozygote is meiotic drive (Section 6.2, Exercises 49–52), where one allele, say A, pushes its way into more than half of the gametes. Another possibility is that the alleles in surviving offspring are not independent. Compare the fraction of heterozygotes produced in the following cases. Compare a case of meiotic drive where 70% of the pollen and 40% of the ovules carry the A allele independently, with a case of nonindependent assortment where an offspring gets an A allele from the pollen with probability 0.7 when the ovule provides an A and gets an A allele from the pollen with probability 0.3 when the ovule provides an a.The ovule provides A with probability 0.5.
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6.5.31 A species of bird comes in three colors: red, blue, and green. Twenty percent are red, 30% are blue, and 50% are green. Females prefer red to blue and blue to green and mate with the best male they find. Females pick the better of the first two males they meet. What is the probability a female mates with a green bird? What did you have to assume about independence?
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6.5.32 A species of bird comes in three colors: red, blue, and green. Twenty percent are red, 30% are blue, and 50% are green. Females prefer red to blue and blue to green and mate with the best male they find. Females pick the better of the first two males they meet. What is the probability a female mates with a blue bird and the probability a female mates with a red bird?
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6.5.33 A small class has only three students. Each comes to class with probability 0.9. Find the probability that all the students come to class and the probability that no students come to class in the following circumstances. The students act independently.
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6.5.35 A small class has only three students. Each comes to class with probability 0.9. Find the probability that all the students come to class and the probability that no students come to class in the following circumstances. Student 2 comes to class with probability 8/9 if student 1 does. Student 3 ignores them.
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6.5.35 A small class has only three students. Each comes to class with probability 0.9. Find the probability that all the students come to class and the probability that no students come to class in the following circumstances. Student 2 comes to class with probability 8/9 if student 1 does. Student 3 ignores them.
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6.5.36 A small class has only three students. Each comes to class with probability 0.9. Find the probability that all the students come to class and the probability that no students come to class in the following circumstances. Student 2 comes to class with probability 8/9 if student 1 does. Student 3 ignores them. Student 3 comes to class with probability 1.0 if both the others come and with probability 1/2 if only one comes. Students 1 and 2 ignore each other.
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6.5.37 In DNA, there are four nucleotides: A, T, C, and G. A pairs with T, and C pairs with G. In many organisms, mutations that change an AT pair into a GC pair are more common than those that change a GC pair into an AT pair. Write down Markov chains describing the following situations. The probability of a switch from AT to GC is 0.002, while a switch from GC to AT occurs with probability 0.001.
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6.5.38 In DNA, there are four nucleotides: A, T, C, and G. A pairs with T, and C pairs with G. In many organisms, mutations that change an AT pair into a GC pair are more common than those that change a GC pair into an AT pair. Write down Markov chains describing the following situations. The probability of a switch from AT to GC is 0.004, while a switch from GC to AT occurs with probability 0.003.
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6.5.39 Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the following situations. It can help to draw a picture. The position of the molecule in one minute is independent of the position in the previous minute.
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6.5.40 Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the following situations. It can help to draw a picture. The molecule leaves a cell with probability 0.1. When it does so, it enters each of the other cells with equal probability. Are the positions independent over time?
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6.5.41 Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the following situations. It can help to draw a picture. Imagine the three cells arranged in a ring. The molecule leaves a cell with probability 0.1, and when it does so, it always moves clockwise.
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6.5.42 Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the following situations. It can help to draw a picture. Imagine the three cells arranged in a line. The molecule makes a given move with probability 0.1. If it is at the end, it moves to the middle. If it is in the middle, it enters the end cells with equal probability.
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6.5.43 From each of the following sets of data, estimate the probability that a molecule that is inside a cell leaves during a given second. Write a discrete-time dynamical system for the probability that the molecule is inside and find the probability it is inside after 3 seconds. How does this probability compare with the fraction of molecules that actually were inside at t =3? Ten molecules start inside a cell. They are first observed outside the cell in the given second. ...
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6.5.44 From each of the following sets of data, estimate the probability that a molecule that is inside a cell leaves during a given second. Write a discrete-time dynamical system for the probability that the molecule is inside and find the probability it is inside after 3 seconds. How does this probability compare with the fraction of molecules that actually were inside at t =3? Ten molecules start inside a cell. They are first observed outside the cell in the given second. ...
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6.5.45 From each of the following sets of data, estimate the probability that a molecule that is inside a cell leaves during a given second, and the probability that it returns. Write a discrete-time dynamical system for the probability that the molecule is inside and find the equilibrium probability. How does the equilibrium probability compare with the fraction of times the molecule actually was inside? One molecule is observed for 20 seconds, and follows ...
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6.5.46 One molecule is observed for 20 seconds, and follows ...
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6.5.47 Figure out a way to simulate 100 offspring from the mechanisms in Section 6.2, Exercises 49 and 50. How close are your results to the mathematically expected results?
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6.5.48 The updating function for the position of a molecule is given by ... where x =1 represents inside, x =0 represents outside, and q is defined in Section 6.1, Exercise 37. Run this system for 50 steps. Based on your data, estimate ... ...Compare these results with what you would expect based on the discrete-time dynamical system.
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