91Ó°ÊÓ

The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations. $$ \begin{array}{lccccc} &&&{\text { Blood Group }} \\ & & \text { O } & \mathbf{A} & \mathbf{B} & \mathbf{A B} \\ \hline {\text { Ethnic Group }} & \mathbf{1} & .082 & .106 & .008 & .004 \\ & \mathbf{2} & .135 & .141 & .018 & .006 \\ & \mathbf{3} & .215 & .200 & .065 & .020 \\ \hline \end{array} $$ Suppose that an individual is randomly selected from the population, and define events by \(A=\\{\) type A selected \(\\}, B=\\{\) type B selected \(\\}\), and \(C=\\{\) ethnic group 3 selected \(\\}\). a. Calculate \(P(A), P(C)\), and \(P(A \cap C)\). b. Calculate both \(P(A \mid C)\) and \(P(C \mid A)\), and explain in context what each of these probabilities represents. c. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?

Step by step solution

01

Calculate P(A)

The probability of selecting an individual with blood type A is found by summing the probabilities across all ethnic groups for blood type A.\[ P(A) = P(A,1) + P(A,2) + P(A,3) = 0.106 + 0.141 + 0.200 = 0.447 \]

02

Calculate P(C)

The probability of selecting an individual from ethnic group 3 is found by summing the probabilities of each blood type within ethnic group 3.\[ P(C) = P(O,3) + P(A,3) + P(B,3) + P(AB,3) = 0.215 + 0.200 + 0.065 + 0.020 = 0.500 \]

03

Calculate P(A ∩ C)

The probability of selecting an individual who has blood type A and is from ethnic group 3 is found directly from the table.\[ P(A \cap C) = P(A,3) = 0.200 \]

04

Calculate P(A | C)

The conditional probability of selecting an individual with blood type A given they are from ethnic group 3 is calculated as:\[ P(A|C) = \frac{P(A \cap C)}{P(C)} = \frac{0.200}{0.500} = 0.4 \]This represents the likelihood that an individual from ethnic group 3 has blood type A.

05

Calculate P(C | A)

The conditional probability of selecting an individual from ethnic group 3 given they have blood type A is calculated as:\[ P(C|A) = \frac{P(A \cap C)}{P(A)} = \frac{0.200}{0.447} \approx 0.448 \]This represents the likelihood that an individual with blood type A is from ethnic group 3.

06

Calculate P(Ethnic Group 1 | Not B)

First, calculate the total probability of not having blood type B.\[ P(\text{not B}) = 1 - [P(B,1) + P(B,2) + P(B,3)] = 1 - (0.008 + 0.018 + 0.065) = 0.909 \]Then calculate the probability of being from ethnic group 1 given the individual does not have blood type B.\[ P(\text{Ethnic Group 1 | Not B}) = \frac{P(\text{Ethnic Group 1 and Not B})}{P(\text{Not B})} \]\[ = \frac{P(O,1) + P(A,1) + P(AB,1)}{0.909} = \frac{0.082 + 0.106 + 0.004}{0.909} = \frac{0.192}{0.909} \approx 0.211 \]So, if an individual does not have type B blood, the probability that they are from ethnic group 1 is approximately 0.211.

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Key Concepts

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

Understanding the Joint Probability Table

A joint probability table is a statistical tool that helps visualize the likelihood of different combinations of outcomes occurring. In this exercise, the table shows the intersection of two variables: ethnic groups and blood types. This kind of table provides a clear view of probability for each possible outcome.

The table is organized as follows:

• Along the top row, you'll see the various blood types (O, A, B, AB).
• The first column lists each ethnic group (1, 2, 3).
• Each cell within the table represents the probability of selecting someone with a specific blood type from a particular ethnic group.

Joint probability, denoted as \( P(X, Y) \), reflects the chance of two specific events happening together. For example, the probability of someone belonging to ethnic group 1 and having blood type O is 0.082. This foundational method allows for deeper analysis, forming the basis for various probability calculations within the exercise.

Ethnic Group Analysis in Probability

Analyzing different ethnic groups' probabilities of having certain blood types involves understanding how these probabilities are spread across each group. In our case, there are three ethnic groups, each contributing to the total population's blood group distribution.

The first step in ethnic group analysis is calculating the probability of selecting an individual from a given group. For example, you calculate \( P(C) \), the probability of selecting someone from ethnic group 3, by summing the probabilities found in the row associated with group 3: \( 0.215 + 0.200 + 0.065 + 0.020 = 0.500 \).

This process ensures you understand each group's share in the population. It’s key to compare these probabilities across groups to observe differences and similarities. This type of analysis not only feeds into finding individual probabilities but also helps in understanding broader socio-demographic trends reflected in the population.

Blood Group Probability and Its Implications

Blood group probability refers to determining the likelihood of certain blood types in the population. Understanding these probabilities helps us find answers to questions about health, compatibility in blood transfusions, or even planning public health strategies.

In our problem, you calculate \( P(A) \), the probability of selecting someone with blood group A, by summing the probabilities for all ethnic groups who have blood group A: \( 0.106 + 0.141 + 0.200 = 0.447 \). This means that 44.7% of the population has blood type A.

Conditional probability takes this a step further, answering questions such as "what's the probability of having blood type A given an individual is from ethnic group 3?" Calculating \( P(A | C) \) answers this, using the formula \( \frac{P(A \cap C)}{P(C)} \). It’s important as it provides insight into how certain characteristics, like blood type, vary within particular subgroups. Understanding blood group probabilities helps in both everyday clinical settings and long-term health planning.