/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 An electronic product contains 4... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 100

An electronic product contains 40 integrated circuits. The probability that any integrated circuit is defective is \(0.01,\) and the integrated circuits are independent. The product operates only if there are no defective integrated circuits. What is the probability that the product operates?

Short Answer

Expert verified
The probability that the product operates is approximately 0.6690.

Step by step solution

01

Understanding the Problem

We have 40 integrated circuits, each with a probability of 0.01 of being defective. They are independent of each other. We need to find out the probability that there are zero defective circuits so that the product operates successfully.
02

Representing with a Probability Model

Since we have multiple independent trials (each integrated circuit working or not working), we can model this situation using the Binomial distribution, where each trial has two outcomes: defective (success) or not defective (failure).
03

Defining Variables and Parameters

For this problem, define the following:- Let each integrated circuit have a probability of defective, \( p = 0.01 \).- The total number of integrated circuits, \( n = 40 \).- We need the probability that exactly 0 circuits are defective.
04

Applying the Binomial Probability Formula

The probability of having exactly \( k = 0 \) defective circuits can be calculated using the Binomial probability mass function:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substitute \( n = 40 \), \( k = 0 \), and \( p = 0.01 \) into the formula:\[ P(X = 0) = \binom{40}{0} (0.01)^0 (0.99)^{40} \]
05

Simplifying the Probability Expression

Since \( \binom{40}{0} = 1 \) and \( (0.01)^0 = 1 \), the probability expression simplifies to:\[ P(X = 0) = 1 \times (0.99)^{40} \]
06

Calculating the Final Probability

Calculate \( (0.99)^{40} \):\[ (0.99)^{40} \approx 0.6690 \]
07

Conclusion

Thus, the probability that the product operates without any defective circuits is approximately 0.6690.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that involves the study of random events. It provides the tools to quantify the likelihood of various outcomes in uncertain scenarios. In the context of the given exercise, probability helps us calculate the likelihood that an electronic product with 40 integrated circuits will operate without any defects.

Understanding probability involves particular concepts and principles, such as events, outcomes, and probability distributions.
  • An **event** is a specific situation that can occur in a random experiment; in this case, a circuit being defective.
  • The **outcomes** are all possible scenarios, such as defective or non-defective circuits.
  • A **probability distribution** assigns probabilities to the different possible outcomes.
When dealing with independent events, probability theory provides the necessary framework for calculating combined probabilities, which is essential in this exercise.
Independent Events
In probability, events are considered independent if the occurrence of one event does not influence the occurrence of another. This is crucial in the context of the integrated circuits problem because it means the status of one circuit (defective or not) doesn鈥檛 affect the others.

To understand independent events, think of them as unconnected trials, where the outcome of one trial has no bearing on the next. In mathematical terms, for two events A and B, they are independent if:
  • The probability of A occurring is unaffected by B:\[P(A|B) = P(A)\]
  • The probability of B occurring is unaffected by A:\[P(B|A) = P(B)\]
In our exercise, each circuit is independent, which lets us use the binomial distribution to find the probability that none of the circuits are defective, hence ensuring the product operates.
Probability Mass Function
A probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. In the context of a binomial distribution, the PMF is used to determine the probability of a given number of successes in a series of independent experiments.

For the binomial distribution, the PMF is expressed as:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where:
  • \( n \) is the number of trials (e.g., number of integrated circuits).
  • \( k \) is the number of successes (in this case, defective circuits).
  • \( p \) is the probability of success on a single trial (probability that a circuit is defective).
  • \( (1-p) \) is the probability of failure (a circuit is not defective) on a single trial.
Applying this to our problem, we find the probability that exactly zero integrated circuits are defective by substituting the given values into the binomial PMF formula, leading to the conclusion that the product operates with a probability of approximately 0.6690.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The probability that an eagle kills a rabbit in a day of hunting is \(10 \%\). Assume that results are independent for each day. (a) What is the distribution of the number of days until a successful hunt? (b) What is the probability that the first successful hunt occurs on day five? (c) What is the expected number of days until a successful hunt? (d) If the eagle can survive up to 10 days without food (it requires a successful hunt on the 10th day ), what is the probability that the eagle is still alive 10 days from now?

Let the random variable \(X\) be equally likely to assume any of the values \(1 / 8,1 / 4,\) or \(3 / 8 .\) Determine the mean and variance of \(X\).

Suppose that a healthcare provider selects 20 patients randomly (without replacement) from among 500 to evaluate adherence to a medication schedule. Suppose that \(10 \%\) of the 500 patients fail to adhere with the schedule. Determine the following: (a) Probability that exactly \(10 \%\) of the patients in the sample fail to adhere. (b) Probability that fewer than \(10 \%\) of the patients in the sample fail to adhere. (c) Probability that more than \(10 \%\) of the patients in the sample fail to adhere. (d) Mean and variance of the number of patients in the sample who fail to adhere.

Each main bearing cap in an engine contains 4 bolts. The bolts are selected at random without replacement from a parts bin that contains 30 bolts from one supplier and 70 bolts from another (a) What is the probability that a main bearing cap contains all bolts from the same supplier? (b) What is the probability that exactly 3 bolts are from the same supplier?

Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is one star per 16 cubic light-years. (a) What is the probability of two or more stars in 16 cubic light-years? (b) How many cubic light-years of space must be studied so that the probability of one or more stars exceeds \(0.95 ?\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.