Wackerly/Mendenhall/Schaeffer Problem 2.19: Assignment of Probabilities

In summary: To justify assigning different probabilities to the simple events, the book would need to explain why they believe certain outcomes are more likely than others. However, without any additional information or context, it seems that all simple events should have an equal probability of occurring.
  • #1
Ackbach
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The Bureau of the Census reports that the median family income for all families in the United States during the year $1991$ was $\$35,353$. That is, half of all American families had incomes exceeding this amount and half had incomes equal to or below this amount (Wright 1992, p. 242). Suppose that three families are surveyed and that each one reveals whether their income exceeded $\$35,353$ in $1991$.

  • List the points in the sample space.
  • Identify the simple events in each of the following events:
    • At least two had incomes exceeding $\$35,353$.
    • Exactly two had incomes exceeding $\$35,353$.
    • Exactly one had income less than or equal to $\$35,353$.
  • Make use of the given information for the median to assign probabilities to the simple events and find $P(A), P(B),$ and $P(C)$.

Answer

  1. Let $E$ mean a family's income exceeds the median, and $B$ mean a family's income was below the median. Then the sample space consists of eight entries: $EEE, EEB, EBE, EBB, BEE, BEB, BBE, BBB$.
  2. We have that \begin{align*} A&=\{EEE, EEB, EBE, BEE\} \\ B&=\{EEB, EBE, BEE\} \\ C&=B. \end{align*}
  3. The probability of each simple event is equal, so $1/8$. $P(A)=1/2$, and $P(B)=P(C)=3/8$.

The problem is that the book's answer is $P(A)=11/16, P(B)=3/8,$ and $P(C)=1/4$. I'm thinking the book assigned different probabilities to the simple events. Why did the book do that? What would be the justification for doing so?
 
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  • #2
Hi Ackbach,

Your answers look right to me. So maybe there is an error in the book.
 

FAQ: Wackerly/Mendenhall/Schaeffer Problem 2.19: Assignment of Probabilities

What is the purpose of Wackerly/Mendenhall/Schaeffer Problem 2.19?

The purpose of this problem is to explore the concept of assigning probabilities in a discrete setting. It involves determining the probabilities of different outcomes in a given scenario, and understanding how to use those probabilities to make predictions and decisions.

What is the background or context of this problem?

This problem is taken from the textbook "Mathematical Statistics with Applications" by Wackerly, Mendenhall, and Schaeffer. It is from Chapter 2, which covers basic concepts and techniques of probability.

What are the main steps involved in solving this problem?

The main steps in solving this problem include:

  1. Identifying the given information and the question being asked.
  2. Defining the sample space for the given scenario.
  3. Assigning probabilities to each possible outcome in the sample space.
  4. Using the assigned probabilities to calculate the desired probability or make a prediction.

What skills or concepts are needed to solve this problem?

To solve this problem, one needs a basic understanding of probability concepts such as sample spaces, events, and probabilities of discrete outcomes. Additionally, knowledge of basic algebra and arithmetic is necessary to perform calculations and solve equations.

How is this problem relevant to real-world applications?

The ability to assign probabilities and make predictions based on those probabilities is crucial in many fields, including statistics, finance, and risk management. This problem provides practice in applying these skills and concepts to a real-world scenario, helping to develop critical thinking and problem-solving abilities.

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