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Chapter 5: Problem 98

Twenty corporations were asked whether or not they provide retirement benefits to their employees. Fourteen of the corporations said they do provide retirement benefits to their employees, and 6 said they do not. Five corporations are randomly selected from these \(20 .\) Find the probability that a., exactly 2 of them provide retirement benefits to their employees. b. none of them provides retirement benefits to their employees. c. at most one of them provides retirement benefits to employees.

Short Answer

Expert verified
The probabilities are calculated using the hypergeometric distribution formula. For part a, the probability that exactly 2 out of the 5 randomly selected corporations provide retirement benefits, the calculated probability needs to be found. For part b, the probability that none of the 5 selected corporations provide retirement benefits, the calculated probability needs to be found. For part c, the probability that at most one of the selected corporations provides retirement benefits, the sum of the calculated probabilities for 0 and 1 will give the result.

Step by step solution

01

Understanding the Hypergeometric Distribution

Hypergeometric distribution is a probability distribution that deals with successes and failures. In this case, 20 corporations can either provide retirement benefits (14 corporations do, which are our 'successes') or not (6 corporations don't, which are our 'failures'). We want to find out the probability of certain combinations in a randomly selected group of 5 corporations.
02

Solving for Part a

To solve this, we want to find the probability that exactly 2 of the randomly selected corporations provide retirement benefits. The formula for hypergeometric probability is \[ P(X=k) = \frac{C(K,k) * C(N-K,n-k)}{C(N,n)} \] where: \[ K = 14 \] (number of corporations that provide benefits), \[ k = 2 \] (number of desired successes), \[ N = 20 \] (total number of corporations), \[ n = 5 \] (number of corporations selected). Using this formula, we can calculate the required probability.
03

Solving for Part b

For part b, we want to find the probability that none of the selected corporations provide retirement benefits. Here, the number of desired successes (k) becomes 0. We again use the same formula as above, but with \[ k = 0 \] this time to calculate the probability.
04

Solving for Part c

In part c, we are to find the probability that at most one of the selected corporations provides retirement benefits. This means we add up the probabilities of having exactly 0 and exactly 1 corporations that provide retirement benefits. So we use the formula mentioned above twice, once with \[ k = 0 \] and then with \[ k = 1 \], and add these two values to get the final probability.

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Key Concepts

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

Probability Calculation
Probability calculation is a method used to determine the likelihood of a specific outcome among all possible outcomes. It plays an essential role in understanding and predicting events in various fields like finance, insurance, and daily life decisions. In our exercise, we're dealing with the hypergeometric distribution, where we calculate the probability of certain combinations of corporations providing retirement benefits.
The probability calculation in hypergeometric distribution involves:
  • Identifying the total number of possible outcomes (the combination of selecting 5 corporations from 20).
  • Determining the number of ways to achieve the desired success (for example, exactly 2 providing retirement).
  • Calculating the ratio of successful outcomes to total possible outcomes using combinatorics.
Each step requires understanding and applying mathematical formulas precisely to estimate the occurrence of any particular event accurately.
Retirement Benefits Statistics
Retirement benefits statistics provide insights into how many corporations offer retirement plans for their employees. These statistics can be vital for job seekers aiming to work at companies with better retirement options. In our exercise, 14 out of 20 corporations offer such benefits, reflecting a fairly high rate of retirement benefits in this sample set.
Such statistics are significant as:
  • They inform decisions for policymakers about mandating retirement options.
  • Help in analyzing economic trends related to employee benefits and compensations.
  • Provide competitive insights to companies on improving their employee benefit packages.
Understanding retirement benefits statistics can also guide individuals in choosing corporations that align with their retirement planning goals.
Combinatorics in Statistics
Combinatorics involves counting, arranging, and calculating the possible combinations and permutations of a set of elements. In statistics, it's essential for calculating probabilities and understanding random processes. The hypergeometric distribution, specifically, uses combinatorial mathematics to determine the probability of a certain number of successes in a sample.
In the context of our problem:
  • We're using combinations to calculate how many ways we can choose 5 corporations from 20.
  • We determine how many of those combinations result in exactly 2 or no corporations providing retirement benefits.
  • We employ the combination formula: \[C(n, k) = \frac{n!}{k!(n-k)!}\]where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
This permits precise calculation of probabilities by analyzing how many successful outcomes occur within the total possible outcomes.
Successes and Failures in Probability
In probability theory, particularly in the hypergeometric distribution, we analyze outcomes as either successes or failures. This framework helps in understanding the probability of various scenarios and is used extensively in real-world applications like quality control, surveys, and risk management.
In our example, a 'success' is a corporation providing retirement benefits (14 successes), while a 'failure' entails not providing such benefits (6 failures). For each selected corporation:
  • Success and failure outcomes allow us to model and predict the probability of choosing corporations with specific characteristics.
  • Analyzing combinations of successes and failures can help in making informed decisions on which corporations might best fit strategic goals or policies.
  • Understanding this balance aids in critical assessments of risk and performance.
Mastering the idea of successes and failures enhances our ability to interpret and predict results across different contexts and domains effectively.

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Most popular questions from this chapter

Although Microsoft Windows is the primary operating system for desktop and laptop PC computers, Microsoft's Windows Phone operating system is installed in only \(1.6 \%\) of smartphones (www. latimes.com/business/la-fi-google- mobile-20110817,0,6230477.story). a. Assuming that \(1.6 \%\) of all current smartphones have Microsoft's Windows Phone operating system, using the binomial formula, find the probability that the number of smartphones in a sample of 80 that have Microsoft's Windows Phone operating system is i. exactly 2 ii. exactly 4 b. Suppose that \(5 \%\) of all current smartphones have Microsoft's Windows Phone operating system. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that in a random sample of 20 smartphones, the number of smartphones that have Microsoft's Windows Phone operating system is i. at most 2 ii. 2 to 3 iii. at least 2

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Suppose the owner of a salvage company is considering raising a sunken ship. If successful, the venture will yield a net profit of \(\$ 10\) million. Otherwise, the owner will lose \(\$ 4\) million. Let \(p\) denote the probability of success for this venture. Assume the owner is willing to take the risk to go ahead with this project provided the expected net profit is at least \(\$ 500,000\). a. If \(p=.40\), find the expected net profit. Will the owner be willing to take the risk with this probability of success? b. What is the smallest value of \(p\) for which the owner will take the risk to undertake this project?

Scott offers you the following game: You will roll two fair dice. If the sum of the two numbers obtained is \(2,3,4,9,10,11\), or 12, Scott will pay you \(\$ 20 .\) However, if the sum of the two numbers is \(5,6,7\), or 8 , you will pay Scott \(\$ 20 .\) Scott points out that you have seven winning numbers and only four losing numbers. Is this game fair to you? Should you accept this offer? Support your conclusion with appropriate calculations.
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