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Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions

If 70% of the cells are young, what is the probability that a cell stains properly? 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.
Find the probability that a cell stains properly. 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
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. Further study of the cell-staining problem (Exercises 29 and 30) reveals that new cells stain properly with probability 0.95,
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? Use Bayes' theorem to compute the above. 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? Use Bayes' theorem to compute the above. 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? Use Bayes' theorem to compute the above. 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? Use Bayes' theorem to compute the above. Say whether the stain is a good indicator of the age of the cell.
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. 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
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. 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.
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. In the above case, 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
Pr(D) = 0.8, Pr(P | D) = 0.95, and Pr(P | N) = 0.1. Compare your results with Exercise 38. In the above case, 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
A plant with genotype Bb is crossed with the offspring from a cross between a BB plant and a Bb plant.Consider a dominant gene where plants with genotype BB or Bb are tall, while plants with genotype bb are short. Determine the probability that a tall plant has genotype Bb when it results from the
A plant with genotype Bb is crossed with the offspring from a cross between a Bb plant and a Bb plant. 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
Two offspring from the cross between a BB plant and a Bb plant are crossed with each other.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
Two tall offspring from the cross between a Bb plant and a Bb plant are crossed with each other.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
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
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}. In the above 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}. In the above 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). In the above cases, find Pr(A ⋂ B), Pr(A | B), and Pr(B | A).
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.) Somebody invents a three-sided die that gives scores of 1, 2,
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). Check whether the above event is independent by checking three equations: Pr(A) = Pr(A |
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.
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.
Show that the multiplication rule (Theorem 6.3) does not work in the following cases.1. For two events A and B that are disjoint, as long as Pr(A) > 0 and Pr(B) > 0.2. For two events A and B where A is a subset of B, as long as Pr(A) > 0 and Pr(B) < 1.
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. Write the information from each of the following two-state Markov chains in terms of conditional probability. Write
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. Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a discrete-time
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. Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a discrete-time dynamical
The caterpillars described in Section 6.2, Exercise 30, where a caterpillar has a probability 0.15 of being taken over by a parasitoid each day and a probability 0.03 of recovering. Write the information from each of the following two-state Markov chains in terms of conditional probability. Write a
Using the probabilities found in Exercise 11, where A = {1, 2}, B = {1, 3}, Pr(A) = 0.4, Pr(B) = 0.6. 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) =
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}. Check whether the above event is independent by checking three equations: Pr(A) = Pr(A |
Using the probabilities found in Exercise 12, where A = {1, 4}, B = {1, 3}, Pr(A) = 0.8, Pr(B) = 0.3. 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) =
1% of people have the disease.Someone comes up with a cut-rate "test" for a disease. This test gives a positive result with probability 0.5 whether or not patient has the disease. In each of the following cases, find the probability of having the disease conditional on a positive test in two
10% of people have the disease.Someone comes up with a cut-rate "test" for a disease. This test gives a positive result with probability 0.5 whether or not patient has the disease. In each of the following cases, find the probability of having the disease conditional on a positive test in two
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.1. What is the probability that both of the molecules remain inside after 1 second (using parameters
What is the probability that both of the molecules remain inside after 2 seconds (using parameters from Section 6.2, Exercise 3)? 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
What is the probability that both of the molecules have moved outside after 2 seconds (using parameters from Section 6.2, Exercise 4)? 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.
Compare a case of meiotic drive where 60% of both pollen and ovules carry the A allele independently, with a case of non independent 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
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}. Check whether the above event is 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 n B) =
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 non independent 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
A species of bird comes in three colors: red, blue, and green. Twenty percent are red, 30% are blue, and 50% are green. Females preferred to blue and blue to green and mate with the best male they find.1. Females pick the better of the first two males they meet. What is the probability a female
Student 2 comes to class with probability 1.0 if student 1 does. Student 3 ignores them. 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 above
Student 2 comes to class with probability 8/9 if student 1 does. Student 3 ignores them.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 above
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. 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
The probability of a switch from AT to GC is 0.002, while a switch from GC to AT occurs with probability 0.001. 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
The probability of a switch from AT to GC is 0.004, while a switch from GC to AT occurs with probability 0.003. 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
The position of the molecule in one minute is independent of the position in the previous minute. Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the above situation. It can help to draw a picture.
The sample space is S = {1, 2, 3, 4}, Pr({l}) = 0.4, Pr({2}) = 0.4, Pr({3}) = 0.1, Pr({4}) = 0.1, A = {1, 2}, and B = {1, 3}. Check whether the above event is 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 n B) =
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? Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the
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. Suppose a molecule is transferred among three cells according to a Markov chain. Write down conditional probabilities to describe the above situation. It can
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. Suppose a molecule is transferred among three cells according to a Markov chain.
Ten molecules start inside a cell. They are first observed outside the cell in the given second. Molecule Time First Observed Outside 1 ...................................... 11 2 ...................................... 1 3 ...................................... 2 4
Ten molecules start inside a cell. They are first observed outside the cell in the given second. Molecule Time First Observed Outside 1 ........................................ 4 2 ........................................ 16 3 ........................................ 14 4
One molecule is observed for 20 seconds, and followsFrom the above set 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
One molecule is observed for 20 seconds, and followsFrom the above set 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
The probability of rolling a 1 followed by a 3. Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Suppose that the results of rolls are independent. Use the multiplication rule to find the above probability. For Information: (Section 6.4, Exercises 9-12)
The probability of rolling three l's in a row. Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Suppose that the results of rolls are independent. Use the multiplication rule to find the above probability. For Information: (Section 6.4, Exercises 9-12)
The probability of rolling two odd values in a row. Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Suppose that the results of rolls are independent. Use the multiplication rule to find the above probability. For Information: (Section 6.4, Exercises
The probability of rolling an odd value followed by an even value. Consider again the three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Suppose that the results of rolls are independent. Use the multiplication rule to find the above probability. For Information: (Section
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.1. The probability of rolling six 3's in a row.2. The
Experiment a. Draw histograms describing the probabilities of the outcomes of four experiments to count the number of mutants in a bacterial cultural.
The histogram in Exercise 9.Using the histogram indicated, estimate the probabilities of the following events.a. The measurement is equal to 7.b. The measurement is less than or equal to 4.c. The measurement is greater than 4.Histogram in Exercise 9
The histogram in Exercise 10.Using the histogram indicated, estimate the probabilities of the following events.a. The measurement is equal to 7.b. The measurement is less than or equal to 4.c. The measurement is greater than 4.Histogram in Exercise 10
The histogram in Exercise 11.Using the histogram indicated, estimate the probabilities of the following events.a. The measurement is equal to 7.b. The measurement is less than or equal to 4.c. The measurement is greater than 4.Histogram in Exercise 11
The histogram in Exercise 12.Using the histogram indicated, estimate the probabilities of the following events.a. The measurement is equal to 7.b. The measurement is less than or equal to 4.c. The measurement is greater than 4.Histogram in Exercise 12
The p.d.f. is f(x) = 2x for 0 ≤ x ≤ 1. For the above p.d.f.'s, Check that the area under the curve is exactly 1. Sketch a graph. Indicate the maximum of the p.d.f., and explain why you are not worried that it is sometimes greater than 1.
The p.d.f. is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. For the above p.d.f.'s, Check that the area under the curve is exactly 1. Sketch a graph. Indicate the maximum of the p.d.f., and explain why you are not worried that it is sometimes greater than 1.
The p.d.f. is h(t) = 1/t for 1 ≤ t ≤ e. For the above p.d.f.'s, Check that the area under the curve is exactly 1. Sketch a graph. Indicate the maximum of the p.d.f., and explain why you are not worried that it is sometimes greater than 1.
Experiment b. Draw histograms describing the probabilities of the outcomes of four experiments to count the number of mutants in a bacterial cultural.
The p.d.f. is g(t) = 6t(1 - t) for 0 ≤ t < 1. For the above p.d.f.'s, a. Check that the area under the curve is exactly 1. b. Sketch a graph. c. Indicate the maximum of the p.d.f., and explain why you are not worried that it is sometimes greater than 1.
The p.d.f. is f(x) = 2x for 0 ≤ x ≤ 1. Find and sketch the c.d.f. associated with the given p.d.f. and check that it increases to a value of 1.
The p.d.f. is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Find and sketch the c.d.f. associated with the given p.d.f. and check that it increases to a value of 1.
The p.d.f. is h(t) = 1/t for 1 ≤ t ≤ e. Find and sketch the c.d.f. associated with the given p.d.f. and check that it increases to a value of 1.
The p.d.f. is g(t) = 6t(1 - t) for 0 ≤ t < 1. Find and sketch the c.d.f. associated with the given p.d.f. and check that it increases to a value of 1.
The p.d.f. is f(x) = 2x for 0 ≤ x ≤ 1. Find the probability that the measurement is between 0.2 and 0.6. Find the probability in two ways: By integrating the given p.d.f. By using the c.d.f. Make sure that your answers match. Shade the given areas on a graph of the p.d.f.
The p.d.f. is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Find the probability that the measurement is between 1.0 and 1.5. Find the probability in two ways: a. By integrating the given p.d.f. b. By using the c.d.f. Make sure that your answers match. Shade the given areas on a graph of the p.d.f.
The p.d.f. is h(t) = 1/t for 1 ≤ t ≤ e. Find the probability that the measurement is between 2.0 and 2.5. Find the probability in two ways: a. By integrating the given p.d.f. b. By using the c.d.f. Make sure that your answers match. Shade the given areas on a graph of the p.d.f.
The p.d.f. is g(t) = 6t(1 - t) for 0 ≤ t ≤ 1. Find the probability that the measurement is between 0.5 and 0.8. Find the probability in two ways: a. By integrating the given p.d.f. b. By using the c.d.f. Make sure that your answers match. Shade the given areas on a graph of the p.d.f.
Sketch the c.d.f. associated with each of the following p.d.f.'s
Experiment c. Draw histograms describing the probabilities of the outcomes of four experiments to count the number of mutants in a bacterial cultural.
Sketch the c.d.f. associated with each of the following p.d.f.'s
Sketch the p.d.f. associated with each of the following c.d.f.'s.
Sketch the p.d.f. associated with each of the following c.d.f.'s.
Sketch the p.d.f. associated with each of the following c.d.f.'s.
Sketch the p.d.f. associated with each of the following c.d.f.'s.
The cells in Section 6.4, Exercise 31, where 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. Draw histograms of the distributions of cell age from the assumptions in the earlier problem. Find and graph the cumulative distribution.
The cells in Section 6.4, Exercise 32, where the cells with age greater than or equal to 3 days have been eliminated from the culture. Draw histograms of the distributions of cell age from the assumptions in the earlier problem. Find and graph the cumulative distribution.
Draw the histogram for experiment a, and find the probability that between 4 and 6 plants (inclusive) are tall.One hundred pairs of plants are crossed, and each pair produces ten offspring. The number of tall offspring is then counted. For the given experiment, draw a histogram of the probability
Draw the histogram for experiment b, and find the probability that between 4 and 6 plants (inclusive) are tall.One hundred pairs of plants are crossed, and each pair produces ten offspring. The number of tall offspring is then counted. For the given experiment, draw a histogram of the probability
Draw the histogram for experiment c, and find the probability that between 4 and 6 plants (inclusive) are tall.One hundred pairs of plants are crossed, and each pair produces ten offspring. The number of tall offspring is then counted. For the given experiment, draw a histogram of the probability
Experiment d. Draw histograms describing the probabilities of the outcomes of four experiments to count the number of mutants in a bacterial cultural.
Draw the histogram for experiment d, and find the probability that between 4 and 6 plants (inclusive) are tall.One hundred pairs of plants are crossed, and each pair produces ten offspring. The number of tall offspring is then counted. For the given experiment, draw a histogram of the probability
Suppose that 100 female birds were tested in each experiment. Find the number out of 200 that mated with each type of male, and convert the results into a probability distribution. An experiment to see which color of male birds female birds prefer is repeated two times. The first time, females mate
Suppose that 100 female birds were tested in the first experiment and 200 females in the second. Find the number out of 300 that mated with each type of male, and convert the results into a probability distribution. An experiment to see which color of male birds female birds prefer is repeated two
Suppose that an equal number of female birds were used in each experiment. Use the law of total probability to find the probability distribution in the combined experiment. An experiment to see which color of male birds female birds prefer is repeated two times. The first time, females mate with
Suppose that three times as many females birds were used in the first experiment. Use the law of total probability to find the probability distribution in the combined experiment. An experiment to see which color of male birds female birds prefer is repeated two times. The first time, females mate
α = 0.5. The p.d.f. for the waiting time X until an event occurs often follows the exponential distribution (to be studied in Section 7.6), with the form g(x) = αe-αx for some positive value of α, defined for x ≥ 0. For each of the following values of α,a. Find the c.d.f.b. Plot the p.d.f.
α = 2.0. The p.d.f. for the waiting time X until an event occurs often follows the exponential distribution (to be studied in Section 7.6), with the form g(x) = αe-αx for some positive value of α, defined for x ≥ 0. For each of the following values of α,a. Find the c.d.f.b. Plot the p.d.f.
The histogram in Exercise 1. Find and sketch the cumulative distribution associated with the histogram from the earlier problem.
The histogram in Exercise 2. Find and sketch the cumulative distribution associated with the histogram from the earlier problem.
The histogram in Exercise 3. Find and sketch the cumulative distribution associated with the histogram from the earlier problem.

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