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Chapter 7: Problem 31

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?

Short Answer

Expert verified
The appropriate sample space for examining three transistors to determine whether they are defective or non-defective is \(\{ddd, ddn, dnd, ndd, nnn, nnd, ndn, dnn\}\), representing all possible combinations of defective (d) and non-defective (n) transistors.

Step by step solution

01

Identify possible outcomes for one transistor

In this experiment, a transistor can be either defective (d) or non-defective (n). So, for a single transistor, we have two possible outcomes: 1. Defective (d) 2. Non-defective (n)
02

Find all possible outcomes for three transistors

Since we have 2 possible outcomes for each of the 3 transistors, we'll need to list all of the combinations of defective and non-defective transistors. In total, there are \(2^3\) = 8 possible outcomes for the three transistors, shown below: 1. ddd: All 3 transistors are defective. 2. ddn: The first 2 transistors are defective, and the third transistor is non-defective. 3. dnd: The first and third transistors are defective, and the second transistor is non-defective. 4. ddn: The first transistor is defective, and the second and third transistors are non-defective. 5. nnn: All 3 transistors are non-defective. 6. nnd: The first 2 transistors are non-defective, and the third transistor is defective. 7. ndn: The first and third transistors are non-defective, and the second transistor is defective. 8. dnn: The first transistor is non-defective, and the second and third transistors are defective.
03

Define the sample space

The sample space is the set of all possible outcomes. In this case, the sample space for the experiment of examining three transistors is: \[\{ddd, ddn, dnd, ndd, nnn, nnd, ndn, dnn\}\]

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Key Concepts

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

Probability Outcomes
In the realm of probability, outcomes are the possible results that can occur in an experiment. For instance, when dealing with transistors, each individual transistor has two potential states: defective (d) or non-defective (n). These are the fundamental outcomes for a single transistor.

When you are looking to find the total possible outcomes in a probability scenario such as the one with three transistors, it is essential to understand that each transistor's state affects the ultimate result. Since each transistor can individually be either defective or non-defective, calculating the total probability outcomes involves considering all possible combinations of these states.

This leads us to exploring a sample space consisting of all the potential outcomes. In this exercise, the sample space is composed of distinct sequences such as `ddd` and `nnn`. It's important to recognize each of these sequences as a unique result in the context of probability.
Defective and Non-defective Analysis
Analyzing the state of transistors as defective or non-defective allows us to explore the quality of a batch of transistors. This type of analysis is essential in quality control.

By defining each transistor as either `d` for defective or `n` for non-defective, we establish a clear framework to assess the quality of multiple transistors together. In a practical analysis, if a batch has more defective transistors, it may indicate problems in manufacturing.

The analysis of defective versus non-defective transistors helps in predicting outcomes in real-world scenarios. For businesses, this insight translates into actionable quality assurance strategies, ensuring that only non-defective products reach the consumer.
Combinations in Probability
Combinations play a significant role in probability, especially when multiple entities are independent but related. In this transistor exercise, combinations help explore different scenarios by arranging defective and non-defective states across the transistors.

To find all possible combinations, we consider that each transistor can independently be either defective or non-defective. Hence, for three transistors, the calculation \((2^3 = 8)\)) is employed. This result represents all unique combinations of states for the transistors.

The combinations are enumerated to generate the sample space, including sequences like `ddn` or `ndn`, each representing a different probability scenario. By understanding how to determine combinations, one learns how to systematically evaluate and predict outcomes in various probability-based situations.

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