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Chapter 3: Problem 107

A group of six software packages available to solve a linear programming problem has been ranked from 1 to 6 (best to worst). An engineering firm, unaware of the rankings, randomly selected and then purchased two of the packages. Let \(Y\) denote the number of packages purchased by the firm that are ranked \(3,4,5,\) or \(6 .\) Give the probability distribution for \(Y\).

Short Answer

Expert verified
The probability distribution for \( Y \) is: \( P(Y=0)=\frac{1}{15}, P(Y=1)=\frac{8}{15}, P(Y=2)=\frac{6}{15} \).

Step by step solution

01

Define the Sample Space

The engineering firm is selecting 2 packages from a total of 6. The total number of ways to choose 2 packages from 6 is given by the combination formula \( \binom{6}{2} = 15 \). This forms our sample space.
02

Identify Favorable Outcomes for Each Y

"Packages 3, 4, 5, and 6 are considered 'successes.' We calculate the number of favorable outcomes for each value of \( Y \):- \( Y = 0 \): Choose 2 from \{1, 2\}. \( \binom{2}{2} = 1 \) way.- \( Y = 1 \): Choose 1 from \{1, 2\} and 1 from \{3, 4, 5, 6\}. \( \binom{2}{1} \cdot \binom{4}{1} = 8 \) ways.- \( Y = 2 \): Choose 2 from \{3, 4, 5, 6\}. \( \binom{4}{2} = 6 \) ways.
03

Calculate Probability for Each Y Value

To find the probabilities, divide the favorable outcomes by the total sample space (15).- Probability for \( Y = 0 \): \( P(Y=0) = \frac{1}{15} \)- Probability for \( Y = 1 \): \( P(Y=1) = \frac{8}{15} \)- Probability for \( Y = 2 \): \( P(Y=2) = \frac{6}{15} \)
04

Construct the Probability Distribution

The probability distribution of \( Y \) is as follows:\[\begin{array}{c|c}Y & P(Y) \ \hline0 & \frac{1}{15} 1 & \frac{8}{15} 2 & \frac{6}{15}\end{array}\]

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Key Concepts

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

Linear Programming
Linear programming is a mathematical method used to determine the best possible outcome in a given mathematical model. The model consists of linear relationships. Linear programming is often used in business and engineering to optimize processes like cost reduction, maximized efficiency, or better resource allocation.

- **Objective Function:** In linear programming, the objective function is a linear equation. It represents the goal of the optimization, such as maximizing profit or minimizing cost.
- **Constraints:** These are linear inequalities or equations which represent the limitations or requirements that must be met in a problem.
- **Feasible Region:** The feasible region is the set of all possible points that satisfy the constraints, within which the optimal solution lies.

Linear programming problems can often be solved using methods like the graphical method (for two variables), the Simplex method, or software packages specifically designed for these tasks. In real-world applications, such as the selection of software packages, linear programming helps in optimal decision-making.
Combinatorics
Combinatorics is a branch of mathematics focused on counting, combinations, and permutations. It helps answer questions like 'How many ways can we select or arrange certain items?' For our exercise, combinatorics is used to determine the number of ways to choose software packages.

- **Combinations:** This is a method where the order of selection doesn鈥檛 matter. The formula used is \( \binom{n}{r} \), which determines the number of ways to choose \( r \) items from \( n \) total items without regard to order.

In the original exercise, by selecting 2 software packages out of 6, combinatorics helps to identify how many different pairs can be formed. For instance, calculating \( \binom{6}{2} = 15 \), gives us the sample space. Similarly, we calculate the number of favorable scenarios for different rankings using combinations, allowing us to determine probabilities.
Sample Space
The sample space in probability is the set of all possible outcomes of a random experiment. When an experiment is conducted, every potential result needs to be considered to understand the entire scope of probabilities.

- **Defining the Sample Space:** In the given exercise, selecting two out of six software packages, we compute a total of 15 possible outcomes using the formula for combinations. This is our sample space, representing every different pair of selections possible.
- **Role in Probability:** The sample space is crucial as it forms the denominator in the fraction used to compute probability. Each particular outcome's probability is calculated by dividing its frequency by the total number of possible outcomes contained in the sample space.

Understanding the sample space gives clarity about the distribution of probabilities across different scenarios, which assists in informed decision-making. Here, it aids in constructing a probability distribution that shows how many of the selected packages belong to a particular ranking, helping in analyzing the probability distribution of \( Y \).

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Most popular questions from this chapter

Suppose that \(Y\) is a discrete random variable with mean \(\mu\) and variance \(\sigma^{2}\) and let \(W=2 Y\) a. Do you expect the mean of \(W\) to be larger than, smaller than, or equal to \(\mu=E(Y) ?\) Why? b. Use Theorem 3.4 to express \(E(W)=E(2 Y)\) in terms of \(\mu=E(Y) .\) Does this result agree with your answer to part (a)? c. Recalling that the variance is a measure of spread or dispersion, do you expect the variance of \(W\) to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y) ?\) Why? d. Use Definition 3.5 and the result in part (b) to show that $$V(W)=E\left\\{[W-E(W)]^{2}\right\\}=E\left[4(Y \mu)^{2}\right]=4 \sigma^{2}$$; that is, \(W=2 Y\) has variance four times that of \(Y\).

An oil prospector will drill a succession of holes in a given area to find a productive well. The probability that he is successful on a given trial is \(.2 .\) a. What is the probability that the third hole drilled is the first to yield a productive well? b. If the prospector can afford to drill at most ten wells, what is the probability that he will fail to find a productive well?

A fire-detection device utilizes three temperature-sensitive cells acting independently of each other in such a manner that any one or more may activate the alarm. Each cell possesses a probability of \(p=.8\) of activating the alarm when the temperature reaches \(100^{\circ}\) Celsius or more. Let Y equal the number of cells activating the alarm when the temperature reaches \(100^{\circ}\). a. Find the probability distribution for \(Y\). b. Find the probability that the alarm will function when the temperature reaches \(100^{\circ}\).

The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of 4 per square yard. Find the probability that a a. 1-square-yard sample will contain at least one imperfection. b. 3-square-yard sample will contain at least one imperfection.

Approximately 4\% of silicon wafers produced by a manufacturer have fewer than two large flaws. If \(Y\), the number of flaws per wafer, has a Poisson distribution, what proportion of the wafers have more than five large flaws? [Hint: Use Table 3, Appendix 3.]
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