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Chapter 3: Problem 92

Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first nondefective engine will be found on the second trial?

Short Answer

Expert verified
The probability is 0.09.

Step by step solution

01

Understand the Problem

We need to find the probability that the first nondefective engine appears on the second trial. Given that 10% of randomly selected engines are defective, this means 90% are nondefective.
02

Define Probability of Events

Define two events: - The probability that an engine is defective (D) is 0.1. - The probability that an engine is nondefective (N) is 0.9.
03

Sequence of Events

We want the first trial to result in a defective engine and the second trial to result in a nondefective engine. We denote this sequence of events as DN.
04

Calculate the Probability of DN

The probability of getting a defective engine (D) first is 0.1, and the probability of getting a nondefective engine (N) second is 0.9. The probability of sequence DN is the product of these probabilities: \( P(DN) = P(D) \times P(N) = 0.1 \times 0.9 \).
05

Solve the Probability

Multiply the probabilities: \( P(DN) = 0.1 \times 0.9 = 0.09 \). This is the probability of the first nondefective engine being found on the second trial.

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Key Concepts

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

Defective Engines
Understanding defective engines is crucial when studying probability in manufacturing contexts. In this particular problem, defective engines are those that do not meet quality standards and may have faults that prevent them from functioning properly. Here, we learn that 10% of engines from the assembly line are defective. This percentage is expressed as a probability of 0.1. Being able to identify defective items in a sequence is essential for predicting outcomes and conducting quality control. In broader scenarios, knowing the proportion of defective products assists companies in improving processes and reducing errors.
Probability Sequence
In probability problems involving sequences, the order of events is a key aspect. For our problem, we were tasked to find the probability that a nondefective engine appears specifically on the second trial. To break down the sequence:
  • The first event in our sequence is drawing a defective engine, which occurs with a probability of 0.1.
  • The second event is drawing a nondefective engine, with a probability of 0.9.
When looking at sequences, the probability of the entire sequence occurring is found by multiplying the probability of each individual event. So, the sequence of having a defective followed by a nondefective engine is calculated by multiplying 0.1 by 0.9. This gives us a total probability for the sequence which is crucial to our answer.
Nondefective Engines
Nondefective engines are those that meet the required standards and function without faults. In the given problem, 90% of the engines are nondefective, leading to a probability of 0.9 for selecting a nondefective engine. The concept of nondefective engines is central in probability scenarios where selecting a working product is desired. In a real-world context, understanding the high probability of nondefective items helps in building confidence in the production process and ensuring customer satisfaction. Moreover, it provides a foundation for calculating probabilities of multiple trials, which often involves sequences of defective and nondefective items.

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Most popular questions from this chapter

The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem?

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Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below certain established subsidy standards. A recent EPA report notes that \(70 \%\) of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates. If five residential subscribers are randomly selected from San Juan, Puerto Rico, find the probability of each of the following events: a. All five qualify for the favorable rates. b. At least four qualify for the favorable rates.
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