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Chapter 5: Problem 108

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
a. The probability exactly 2 of them provide retirement benefits to their employees is approximately 0.1323.\n b. The probability none of them provides retirement benefits to their employees is approximately 0.00243.\n c. The probability that at most one of them provides retirement benefits to their employees is approximately 0.03078.

Step by step solution

01

Identify parameters

Our parameters are \(n = 5\) (sample size), \(p = 0.7\) (probability of success), and \(q = 0.3\) (probability of failure).
02

Calculate probability for part a

For part a, we have 2 successes. This means \(x = 2\). So, we plug the values into the binomial probability formula: \(P(2;5,0.7) = C(5, 2) * 0.7^2 * 0.3^3\), which calculates to approximately 0.1323.
03

Calculate probability for part b

For part b, we have 0 successes. So, \(x = 0\). We put the values into the binomial probability formula: \(P(0;5,0.7) = C(5, 0) * 0.7^0 * 0.3^5\), giving approximately 0.00243.
04

Calculate probability for part c

For part c, we have at most 1 success. This means we have to calculate for \(x = 0\) and \(x = 1\), and then add the probabilities. So, \(P(1;5,0.7) = C(5, 1) * 0.7^1 * 0.3^4 = 0.02835\). Add the probabilities for x = 0 and x = 1 to get the answer for part c, \(0.00243 + 0.02835 = 0.03078\).

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Key Concepts

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

Probability of Success
In the context of binomial distribution, the probability of success refers to the likelihood of a desired outcome occurring in a trial. For instance, when surveying corporations about retirement benefits, a "success" would mean that a corporation offers such benefits.

To determine the probability of success, you divide the number of successes by the total number of trials. In our example, out of 20 corporations, 14 provide retirement benefits. Therefore, the probability of success, denoted as \(p\), is \(\frac{14}{20} = 0.7\). This probability is crucial because it represents the rate at which we expect a corporation to offer benefits within the sample.
Binomial Probability Formula
The binomial probability formula allows us to calculate the probability of obtaining a specific number of successes in a fixed number of trials, given a known probability of success in each trial.

The formula is given by:
  • \(P(x;n,p) = C(n, x) \cdot p^x \cdot (1-p)^{n-x}\)
In this formula:
  • \(C(n, x)\) represents the number of combinations for choosing \(x\) successes out of \(n\) trials.
  • \(p\) is the probability of success in each trial.
  • \((1-p)\) is the probability of failure.
This formula is vital because it allows you to calculate the likelihood of any given number of successes based on the binomial distribution parameters.
Sample Size
Sample size, in the context of binomial distribution, refers to the number of trials conducted in an experiment or survey. This is denoted by \(n\). In our example, the sample size is \(5\) because we are selecting 5 corporations.

The sample size is critical as it affects the distribution's behavior; a larger sample size can lead to more accurate probability estimates. However, in a binomial distribution, each trial is considered independent, meaning that selecting one corporation doesn't influence the selection of another.
Probability Calculation
Calculating probabilities using the binomial distribution involves substituting the correct values into the binomial probability formula.

Consider calculating the probability that exactly 2 out of 5 corporations provide benefits. Using the binomial formula:
  • \(P(2;5,0.7) = C(5, 2) \cdot (0.7)^2 \cdot (0.3)^3\)
  • \(C(5, 2)\) calculates the ways to choose 2 successes from 5 trials.
  • \((0.7)^2\) represents the probability of success for 2 corporations.
  • \((0.3)^3\) accounts for the probability of failure for the remaining 3 corporations.
Performing the calculations provides a probability of about 0.1323. This systematic approach helps us understand how likely certain outcomes are across various scenarios.

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Most popular questions from this chapter

Six jurors are to be selected from a pool of 20 potential candidates to hear a civil case involving a lawsuit between two families. Unknown to the judge or any of the attorneys, 4 of the 20 prospective jurors are potentially prejudiced by being acquainted with one or more of the litigants. They will not disclose this during the jury selection process. If 6 jurors are selected at random from this group of 20, find the probability that the number of potentially prejudiced jurors among the 6 selected jurors is a. exactly \(\overline{1}\) b. none c. at most 2

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