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Chapter 4: Problem 83

Forty-seven employees in an office wear eyeglasses. Thirty-one have single- vision correction, and 16 wear bifocals. If two employees are selected at random from this group, what is the probability that both of them wear bifocals? What is the probability that both have single-vision correction?

Short Answer

Expert verified
The probability for both employees to wear bifocals is \( \frac{{16}}{{47}} \times \frac{{15}}{{46}} \) and for single-vision correction is \( \frac{{31}}{{47}} \times \frac{{30}}{{46}} \).

Step by step solution

01

Calculate Total Probability and Individual Probabilities

First, determine the total number of employees, which is 47. Then, determine the number of employees who wear bifocals (16) and single-vision correction (31). The individual probabilities for selecting a bifocal wearer or a single-vision correction wearer are calculated as follows: \[ P(\text{{Bifocals}}) = \frac{{\text{{Number of bifocal wearers}}}}{{\text{{Total number of employees}}}} = \frac{{16}}{{47}} \] and \[ P(\text{{Single-vision}}) = \frac{{\text{{Number of single-vision wearers}}}}{{\text{{Total number of employees}}}} = \frac{{31}}{{47}} \].
02

Calculate Joint Probabilities

Next, calculate the joint probability of selecting two employees with bifocals and two with single-vision correction. The joint probability can be calculated using the multiplication of individual probabilities for independent events (since the probability of the second selection depends on the outcome of the first, and vice versa). \[ P(\text{{2 Bifocals}}) = P(\text{{Bifocals}}) \times P(\text{{Bifocals}}) |_{\text{{after 1st selection}}} = \frac{{16}}{{47}} \times \frac{{15}}{{46}} \] and \[ P(\text{{2 Single-vision}}) = P(\text{{Single-vision}}) \times P(\text{{Single-vision}}) |_{\text{{after 1st selection}}} = \frac{{31}}{{47}} \times \frac{{30}}{{46}} \].

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Key Concepts

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

Joint Probability
Joint probability refers to the likelihood of two events happening at the same time. In the context of our exercise, it involves determining the probability of selecting two specific types of employees consecutively, such as two bifocal wearers or two single-vision correction wearers. This calculation involves multiplying the probability of selecting the first individual with the probability of selecting the second individual, given the first event has already occurred. By understanding joint probability, we can better grasp scenarios where multiple criteria or outcomes are being assessed together.
Independent Events
While calculating probabilities, it's crucial to determine whether events are independent. Independent events mean the occurrence of one event does not affect the likelihood of the other. In our case, however, choosing one employee affects the pool from which the next is selected, making these events dependent. Thus, the probability of selecting a bifocal wearer for the second draw depends on the first selection having been a bifocal wearer as well. This illustrates how event dependency impacts joint probability calculations, reinforcing why we adjust denominators when computing probabilities for sequential events.
Single-vision Correction
Single-vision correction refers to eyeglasses designed for people who need assistance seeing clearly at one distance, either near or far. These glasses are prevalent, as shown by 31 employees wearing them in the exercise. Understanding this type of correction is critical for probability calculation, particularly when considering the likelihood of drawing two single-vision correction wearers successively. Calculating this probability involves considering how many wear single-vision correction initially and how this number changes with the first selection.
Bifocals
Bifocals are specialized glasses designed for individuals who need assistance seeing clearly at multiple distances. They boast separate lens sections for near and far vision tasks. In the given exercise, 16 employees are bifocal wearers. Calculating the probability of two randomly selected employees wearing bifocals involves understanding how the bifocal group's size changes after one selection. Such understanding is pivotal when dealing with dependent events in probability studies. By grasping these nuances, students can more accurately compute probabilities and understand real-world applications.
Probability Calculation
Probability calculation is the process of determining how likely an event is to occur. For our exercise, it means determining the chances of selecting two consecutive individuals either with single-vision correction or bifocals. Calculations involve fractions that describe different scenarios:
  • Choosing the first item with a specific lens type.
  • Adjusting the selection pool for the second item.
For both bifocals and single-vision, the probability changes after the first pick, affecting the overall chances. Always remember that calculating probabilities in sequence requires an understanding of how each pick alters the remaining pool of options.

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Most popular questions from this chapter

The probability that a farmer is in debt is 80 . What is the probability that three randomly selected farmers are all in debt? Assume independence of events.

Five hundred employees were selected from a city's large private companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$ \begin{array}{lcc} &{\text { Have Retirement Benefits }} \\ \hline { 2 - 3 } & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array} $$ a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events "woman" and "yes" ii. Probability of the intersection of events "no" and "man" b. Mention what other joint probabilities you can calculate for this table and then find them. You may draw a tree diagram to find these probabilities.

A random sample of 80 lawyers was taken, and they were asked if they are in favor of or against capital punishment. The following table gives the two-way classification of their responses. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Favors Capital } \\ \text { Punishment } \end{array} & \begin{array}{c} \text { Opposes Capital } \\ \text { Punishment } \end{array} \\ \hline \text { Male } & 32 & 24 \\ \text { Female } & 13 & 11 \\ \hline \end{array} $$ a. If one lawyer is randomly selected from this group, find the probability that this lawyer i. favors capital punishment ii. is a female iii. opposes capital punishment given that the lawyer is a female iv. is a male given that he favors capital punishment \(\mathrm{v}\). is a female and favors capital punishment vi. opposes capital punishment or is a male b. Are the events "female" and "opposes capital punishment" independent? Are they mutually exclusive? Explain why or why not.

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \\ \hline \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of major iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.

How is the addition rule of probability for two mutually exclusive events different from the rule for two mutually nonexclusive events?
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