Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

The sample spaces are large and you should use the counting principles discussed in Section 9.5. A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four of these units, and because all are packaged identically, the selection will be random. What is the probability that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good?

Step-by-Step Solution

Answer

An error has occurred in the calculation. Probabilities cannot be greater than 1. The steps followed for (a) and (b) are correct; however, the calculations for (c) are not. Please revise the problem and calculations.

Step 1: Identify key values

In this problem, there are 12 microwave ovens, of which 9 are good and 3 are defective. The vending company orders four units.

Step 2: Calculate the total number of combinations

The total number of possible combinations when choosing 4 ovens out of 12 is given by the combination formula \( C(n, m) = \frac{n!}{m!(n-m)!} \). So \( C(12, 4) = 495 \) combinations.

Step 3: (a) All four units are good

The total number of ways to choose 4 good ovens out of 9 is \( C(9, 4) = 126 \). Therefore, the probability is \( \frac{126}{495} = 0.2545 \) approximately.

Step 4: (b) Exactly two units are good

The total number of ways to choose 2 good ovens out of 9 and 2 defective ovens out of 3 is \( C(9, 2) * C(3, 2) = 252 \). Therefore, the probability is \( \frac{252}{495} = 0.5091 \) approximately.

Step 5: (c) At least two units are good

The total number of ways to choose at least two good ovens (2, 3, or 4) and the rest defective is \( C(9, 2) * C(3, 2) + C(9, 3) * C(3, 1) + C(9, 4) = 126 + 252 + 126 = 504 \). Therefore, the probability is \( \frac{504}{495} = 1.0182 \). This is greater than 1 because there are more ways than the total number of combinations which indicates there has been an error.

Key Concepts

Combination formula

In probability and statistics, the combination formula is a fundamental tool used to determine the number of possible ways to choose a certain number of items from a larger set without regard to order. The combination formula is given by:

• \( C(n, m) = \frac{n!}{m!(n-m)!} \)

Here, \(n\) represents the total number of items to choose from, \(m\) is the number of items to be chosen, and \(!\) denotes factorial, which means multiplying a series of descending natural numbers.
For example, in the context of choosing 4 microwave ovens out of 12, we use the combination formula \( C(12, 4) \), which yields 495 possible combinations. This helps us calculate probabilities by understanding all possible selection outcomes without considering order.

Counting principles

Counting principles are fundamental rules that help us systematically count the number of ways certain arrangements or selections can occur. These principles include:

• The Rule of Product (Multiplication Principle): This states that if there are \(a\) ways to do one thing and \(b\) ways to do another, there are \(a \, \times \, b\) ways to do both.
• The Addition Principle: If one activity can occur in \(a\) different ways and another can occur in \(b\) different ways, totaling \(a + b\), provided these activities do not overlap.

These principles are essential in determining the total combinations or arrangements in complex problems, like choosing different numbers of good and defective ovens from the shipment. For instance, determining how to choose 2 good units and 2 defective units uses these principles by multiplying the possible ways of each selection.

Defective units

Defective units refer to items in a sample or production batch that do not meet quality standards. In the given exercise, out of 12 microwave ovens, three units are defective, meaning they do not operate correctly or are not up to specified standards. Understanding the impact of defective units is crucial in scenarios like:

• Quality assurance during manufacturing and supply.
• Accurate computation of probability in mixed-quality lots.
• Making informed decisions on product selection and risk management.

In probability, the presence of defective units adjusts calculations, such as when we calculate the probability of picking no defective units or exactly two defective units from a batch, influencing the final probability outcomes.

Sample spaces

In the study of probability, a sample space is the set of all possible outcomes of a random experiment. When dealing with sample spaces, it's essential to consider both the arrangement and selection of items. For the shipment of microwave ovens:

• The sample space of all possible combinations when picking four units from the shipment is represented by \( C(12, 4) = 495 \).
• Each subset of 4 units is a potential outcome or sample in this experiment.
• The sample space helps us determine the probability of each event by dividing the favorable outcomes by the total number of outcomes.

Understanding the complete sample space is crucial for correctly calculating probabilities and ensuring every possible outcome is considered, allowing for more accurate predictions and analyses.