Solutions Modeling Dynamics of Life 3ed Adler - Chapter 6.4

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.
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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.
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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.
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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.
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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}.
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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}.
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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}.
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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}.
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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.)
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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.
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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.
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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.
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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).
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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).
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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).
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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).
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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.
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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.
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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?
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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?
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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.
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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.
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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?
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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?
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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?
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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?
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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?
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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?
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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?
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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?
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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?
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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?
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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.
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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.
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