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Chapter 2: Problem 74

The proportions of blood phenotypes in the U.S. population are as follows: \(\begin{array}{lccc}\mathrm{A} & \mathrm{B} & \mathrm{AB} & \mathrm{O} \\ .40 & .11 & .04 & .45\end{array}\) Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individuals match?

Short Answer

Expert verified
The probability that both have phenotype O is 0.2025. The probability that both have the same phenotype is 0.3762.

Step by step solution

01

Understanding the Problem

We need to determine the probability of certain blood phenotypes for two randomly chosen individuals from the U.S. population. The first task is to calculate the probability that both have blood phenotype O, and then the probability that both have the same phenotype, regardless of type.
02

Calculate Probability for Both O

The probability of a single individual having phenotype O is given as 0.45. Since the phenotypes for the two individuals are independent, use the multiplication rule for independent events to find the probability that both are O. \[ P( ext{both O}) = P(O) imes P(O) = 0.45 imes 0.45 = 0.2025 \]
03

Calculate Probability of Matching Phenotypes

Calculate the probability that both individuals have the same phenotype. This includes them both being A, both being B, both being AB, or both being O. Use the formula for each phenotype separately and add them:- Both A: \( P( ext{both A}) = 0.40 imes 0.40 = 0.16 \)- Both B: \( P( ext{both B}) = 0.11 imes 0.11 = 0.0121 \)- Both AB: \( P( ext{both AB}) = 0.04 imes 0.04 = 0.0016 \)- Both O: \( P( ext{both O}) = 0.45 imes 0.45 = 0.2025 \)Add these probabilities:\[ P( ext{matching phenotypes}) = 0.16 + 0.0121 + 0.0016 + 0.2025 = 0.3762 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Phenotype Distribution
Blood phenotypes refer to the different types of blood groups present within a population. In the U.S., the distribution of blood phenotypes consists of Type A, Type B, Type AB, and Type O. Each of these phenotypes occurs at different frequencies. For example, Type O is the most common at 45%, followed by Type A at 40%, Type B at 11%, and Type AB at 4%.
Understanding these distributions is crucial when calculating probabilities related to blood types. If you know the proportion of each phenotype, you can predict the likelihood of an individual having a particular type based on these population statistics. Knowing these details is essential for tasks such as medical planning and resource allocation.
Independent Events
Independent events are situations where the outcome of one event does not affect the outcome of another. This is a key concept in probability, often assumed when considering events involving multiple individuals in a large population.
In the context of blood phenotype distribution, assuming that the blood type of one person does not influence another means the chances remain consistent. If an individual has a particular blood type, it doesn't alter the statistical likelihood of the next person having any blood type. This assumption simplifies calculations and makes predicting outcomes using statistical models straightforward.
  • Each choice is made without regard to others.
  • Results from different individuals remain random.
Recognizing when events are independent allows for easier application of probability rules, such as those used in this exercise.
Multiplication Rule
The multiplication rule is a fundamental principle in probability theory. It states that the probability of two independent events both happening is the product of their individual probabilities. In other words, for two independent events A and B, the probability of both A and B occurring is given by:
  • \(P(A \text{ and } B) = P(A) \times P(B)\).
This principle is directly applied when calculating the probability of two individuals possessing the same blood type. Since each person's blood type is an independent event, you'd multiply the probability of each outcome to find the combined probability as was done when finding the likelihood of both individuals having Type O blood type.
Matching Phenotypes Probability
Matching phenotypes probability refers to calculating the chance that two randomly selected individuals have the same blood phenotype. This requires considering all possible matches, not just a specific type.
The solution involves computing probabilities for all scenarios where the two individuals have the same blood type. By multiplying the probability for each blood type pair (e.g., both A, both B) and summing them, we obtain the overall probability that the phenotypes match.
  • Both A: \(0.40 \times 0.40\)
  • Both B: \(0.11 \times 0.11\)
  • Both AB: \(0.04 \times 0.04\)
  • Both O: \(0.45 \times 0.45\)
After adding these products, we find the total probability for matching phenotypes at 0.3762. This comprehensive approach covers all possibilities and provides the complete picture of the likelihood of individuals sharing blood types.

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Most popular questions from this chapter

The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is .4, the analogous probability for the second signal is \(.5\), and the probability that he must stop at at least one of the two signals is .6. What is the probability that he must stop a. At both signals? b. At the first signal but not at the second one? c. At exactly one signal?

An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let \(A\) be the event that the Asian project is successful and \(B\) be the event that the European project is successful. Suppose that \(A\) and \(B\) are independent events with \(P(A)=.4\) and \(P(B)=.7\). a. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning. b. What is the probability that at least one of the two projects will be successful? c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?

An engineering construction firm is currently working on power plants at three different sites. Let \(A_{i}\) denote the event that the plant at site \(i\) is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \(A_{1}, A_{2}\), and \(A_{3}\), draw a Venn diagram, and shade the region corresponding to each one. a. At least one plant is completed by the contract date. b. All plants are completed by the contract date. c. Only the plant at site 1 is completed by the contract date. d. Exactly one plant is completed by the contract date. e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is \(.9\), the probability that at least one of the two components does so is \(.96\), and the probability that both components do so is .75. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also?

Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events \(A_{1}\), \(\mathrm{A}_{2}\), and \(\mathrm{A}_{3}\) by \(A_{1}=\) likes vehicle \(\\# 1 \quad A_{2}=\) likes vehicle \(\\# 2\) \(A_{3}=\) likes vehicle \(\\# 3\) Suppose that \(P\left(A_{1}\right)=.55, P\left(A_{2}\right)=.65, P\left(A_{3}\right)=.70\), \(P\left(A_{1} \cup A_{2}\right)=.80, P\left(A_{2} \cap A_{3}\right)=.40\), and \(P\left(A_{1} \cup A_{2} \cup A_{3}\right)=.88\). a. What is the probability that the individual likes both vehicle #1 and vehicle #2? b. Determine and interpret \(P\left(A_{2} \mid A_{3}\right)\). c. Are \(\mathrm{A}_{2}\) and \(\mathrm{A}_{3}\) independent events? Answer in two different ways. d. If you learn that the individual did not like vehicle #1, what now is the probability that he/she liked at least one of the other two vehicles?
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