Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A sample of three transistors taken from a local electronics store was examined to determine whether the transistors were defective \((d)\) or nondefective \((n) .\) What is an appropriate sample space for this experiment?

Step-by-Step Solution

Answer

An appropriate sample space for this experiment is \(\{nnn, ndd, dnd, ddn, nnd, ndn, dnn, ddd\}\), representing all possible combinations of defective \((d)\) and nondefective \((n)\) transistors among the three selected.

Step 1: Identify the fundamental outcome unit

In this experiment, each transistor can be either defective \((d)\) or nondefective \((n)\). So, the fundamental outcome unit consists of these two possibilities.

Step 2: Construct the sample space by considering all the possible outcomes for three transistors

Let's consider all possible outcomes for the three transistors selected: 1. All three transistors are nondefective: (nnn) 2. The first transistor is nondefective, and the other two are defective: (ndd) 3. The second transistor is nondefective, and the other two are defective: (dnd) 4. The third transistor is nondefective, and the other two are defective: (ddn) 5. Two nondefective transistors and one defective transistor: (nnd), (ndn), (dnn) 6. All three transistors are defective: (ddd)

Step 3: Write down the sample space

Now that we have covered all possible outcomes, we can write down the sample space for this experiment. The sample space will consist of all possible outcomes for selecting three transistors which are either defective or nondefective: Sample Space = {nnn, ndd, dnd, ddn, nnd, ndn, dnn, ddd}

Key Concepts

Defective and Nondefective Transistors

In any given batch of transistors, each unit can fall into one of two categories: defective (d) or nondefective (n). This distinction is crucial for quality control and reliability purposes in manufacturing and electronics.

• Defective transistors are those that fail to meet performance criteria and may malfunction, compromising the integrity of the electronic devices they are used in.
• Nondefective transistors operate as intended, ensuring the proper function of electronic components.

When conducting an experiment that involves identifying defective and nondefective transistors, one must carefully label and record each transistor’s status. This task might seem simple in a sample of three, like in our exercise, but understanding this binary classification allows for more complex analyses in larger datasets. It's a fundamental principle in statistical quality control that aids in improving manufacturing processes by identifying and reducing defects.

Probability Outcomes

In the realm of probability, outcomes are the distinct results that can occur from a process or experiment. When dealing with transistors, the outcomes revolve around whether each transistor is defective or nondefective.

• Each set of outcomes, like (nnn) or (dnd), represents a possible scenario from the sample space.
• The concept of an "outcome" is important because it allows us to analyze all potential results quantitatively.

In this exercise, understanding probability outcomes helps in calculating the likelihood of different combinations of defective and nondefective transistors. This approach is foundational in determining probabilities in larger datasets or more complex experiments involving uncertainty and risk assessment. For example, if you find more (nnn) outcomes than (ddd), it indicates a higher probability of transistors being nondefective, signaling a satisfactory manufacturing process.

Fundamental Outcomes in Experiments

Fundamental outcomes form the building blocks of any probability experiment. For the case of transistors, the fundamental outcome for each transistor is whether it is defective (d) or nondefective (n). From these primary results, we can construct the complete set of possible combinations, known as the sample space.

• For any small set of events, like three transistors, listing all fundamental outcomes allows us to see all potential scenarios clearly. This clarity is essential for accurately defining the experiment's sample space.
• The comprehensive list—such as {nnn, ndd, dnd, ddn, nnd, ndn, dnn, ddd}—is inclusive of every possible way the transistors could be defective or not.

Understanding these fundamental outcomes and their combinations lends insight into how often certain configurations might appear in real-world scenarios. It ensures we don't overlook any possibilities when analyzing experimental results, which in turn supports accurate probability analysis and decision-making.