probability problems, first two days

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1 ph 141 probability problems, first two days from Moore, McCabe and Craig, 8th ed Exercise and sleep. Suppose that 40 % of adults get enough sleep, 46 % get enough exercise, and 24 % do both. Find the probabilities of the following events: (a) enough sleep and not enough exercise (b) not enough sleep and enough exercise (c) not enough sleep and not enough exercise (d) For each of parts (a), (b) and (c), state the rule that you used to find your answer Exercise and sleep. Refer to the previous exercise. Draw a Venn diagram showing the probabilities for exercise and sleep Distribution of blood types. All human blood can be ABO-typed as one of O, A, B, or AB, but the distribution of types varies a bit among groups of people. Here is the distribution of blood types for a randomly chosen person in the United States: Blood type A B AB O U.S. probability ? 0.44 (a) What is the probability of type AB blood in the United States? (b) Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen person from the United States can donate blood to Maria? 4.39 Universal blood donors. People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of type O-negative blood. Only 7 % of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least 1 of them is a universal donor? 1

2 4.35 Rh blood types. Human blood is typed as O, A, B, or AB and also as Rh-positive or Rh-negative. ABO type and Rh-factor are independent because they are governed by different genes. In the American population, 84 % of people are Rh-positive. Use the information about ABO-type in exercise 4.29 to give the probability distribution of blood type (ABO and Rh) for a randomly chosen American Blood types in Ireland. The distribution of blood types in Ireland differs from the U.S. distribution given in [exercise 4.29]: Blood type A B AB O Ireland probability Choose a person from the United States and a person from Ireland at random, independently of each other. What is the probability that both have type O blood? What is the probability that both have the same blood type? Mendelian inheritance. Some traits of plants and animals depend on inheritance of a single gene. This is called Mendelian inheritance, after Gregor Mendel ( ). Exercises 4.42 to 4.45 are based on the following information about Mendelian inheritance of blood type. Each of us has an ABO blood type, which describes whether two characteristics called A and B are present. Every human being has two blood type alleles (gene forms), one inherited from our mother and one from our father. Each of these alleles can be A, B, or O. Which two we inherit determines our blood type. Here is a table that shows what our blood type is for each combination of two alleles. Alleles inherited A and A A and B A and O B and B B and O O and O Blood type A AB A B B O 2

3 We inherent each of a parent s two alleles with probability 05. We inherit independently from our mother and our father Blood types of children. Hannah and Jacob both have alleles A and B. (a) What blood types can their children have? (b) What is the probability that their next child has each of these blood types? 4.43 Parents with alleles B and O. Nancy and David (a) What blood types can their children have? (b) What is the probability that their next child has each of these blood types? 4.44 Two children. Jennifer has alleles A and O. José has alleles A and B. They have two children. [ML: not identical twins!] What is the probability that both children have blood type A? What is the probability that both children have the same blood type? 4.45 Three children. Jasmine has alleles A and O. Joshua has alleles B and O. (a) What is the probability that a child of these parents has blood type O? (b) If Jasmine and Joshua have three children [ML: again, not identical twins!] what is the probability that all three have blood type O? what is the probability that the first child has blood type O and the next two do not? 3

4 4.115 Attendance at two-year and four-year colleges. In a large national population of college students. 61 % attend four-year institutions and the rest attend two-year institutions. Males make up 44 % of the students in the four-year institutions and 41 % of the students in the twoyear institutions. (a) Find the four probabilities for each combination of gender and type of institution in the following table. Be sure the probabilities sum to 1. Men Women Four-year institution Two-year institution (b) Consider randomly selecting a female student from this population. What is the probability that she attends a four-year institution? Draw a tree diagram. Refer to the previous exercise. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the type of institution attended and then identify the gender of the student Draw a different tree diagram for the same setting. Refer to the previous two exercises. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the gender of the student and then the type of institution attended. Explain why the probabilities in this tree diagram are different from the those you used in the previous exercise [exercise 4.116]? 4.33 Education levels of young adults. Choose a young adult (age 25 to 34 years) at random. The probability is 0.12 that the person chosen did not complete high school, 0.31 that the person has a high school education, and 0.29 that the person has at least a bachelor s degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor s degree? (b) What is the probability that a randomly chosen young adult has at least a high school education? 4

5 4.38 PINs. The personal identification numbers (PINs) for automatic teller machines usually consist of four digits. You notice that most of your PINs have at least one 0, and you wonder if the issuers use lots of 0s to make the numbers easy to remember. Suppose that PINs are assigned at random, so that all four-digit numbers are assigned at random, so that all four-digit numbers are equally likely. (a) How many possible PINs are there? (b) What is the probability that a PIN assigned at random has at least one 0? Education and income. Call a household prosperous if its income exceeds $100,000. Call a household educated of the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated. According to the Current Population Survey, P(A) = 0.138, P(B) = 0.261, and P(A and B) = What is the probability P(A or B) that the household selected is either prosperous or educated? Find a conditional probability. In the setting of the previous exercise, what is the conditional probability that a household is prosperous, given that it is educated? Explain why your result shows that events A and B are not independent Draw a Venn diagram. Draw a Venn diagram that shows the relation between the events A and B in exercise Indicate each of the following events on your diagram and use the information in exercise to calculate the probability of each event. Finally, describe in words what each event is. (a) A and B (b) A c and B (c) A and B c (d) A c and B c Sales of cars and light trucks. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. In a recent year, 69 % of vehicles sold were light trucks, 78 % were domestic, and 55 % were domestic light trucks. Let A be the event that a vehicle is a car and B the event that it is imported. Write each of the following events in set notation and give its probability. (a) The vehicle is a light truck. (b) The vehicle is an imported car. 5

6 Genetic counseling. Conditional probabilities and Bayes s rule are a basis for counseling people who may have genetic defects that can be passed on to their children. Exercises to concern genetic counseling settings Albinism. People with albinism have little pigment in their skin, hair, and eyes. The gene that governs albinism has two forms (called alleles), which we denote by a and A. Each person has a pair of these genes, one inherited from each parent. A child inherits one of each parent s two alleles independently with probability 0.5. Albinism is a recessive trait, so a person is albino only if the inherited pair is aa. (a) Beth s parents are not albino but she has an albino brother. This implies that both of Beth s parents have type Aa. Why? (b) Which of the types aa, Aa, AA could a child of Beth s parents have? What is the probability of each type? (c) Beth is not albino. What are the conditional probabilities for Beth s possible types, this fact? given Find some conditional probabilities. Beth knows the probabilities of her genetic types from part (c) of the previous exercise. She marries Bob, who is albino. Bob s genetic type must be aa. (a) What is the conditional probability that a child of Beth and Bob is non-albino if Beth has type Aa? What is the conditional probability of a non-albino child if Beth has type AA? (b) Beth and Bob s first child is non-albino. What is the conditional probability that Beth is a carrier, type Aa? 6