A company producing fine crystal goblets has, in the past, experienced a defect rate of 10% in its goblets. Such goblets are designated as "seconds". Assume defective goblets are independent. a) With what probability will there be one "second" among six randomly selected goblets? b) with what probability will there be at least four "seconds" among twelve randomly selected goblets? c) What is the expected number of "seconds" among 50 goblets? What is the associated variance? Hint: Binomial Distribution
A company producing fine crystal goblets has, in the past, experienced a defect rate of 10% in its goblets. Such goblets are designated as "seconds". Assume defective goblets are independent. a) With what probability will there be one "second" among six randomly selected goblets? b) with what probability will there be at least four "seconds" among twelve randomly selected goblets? c) What is the expected number of "seconds" among 50 goblets? What is the associated variance? Hint: Binomial Distribution
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Analysis of Defective Crystal Goblets Using Binomial Distribution
#### Problem Overview
A company producing fine crystal goblets has, in the past, experienced a defect rate of 10% in its goblets. Such goblets are designated as "seconds". Assume defective goblets are independent.
#### Questions to be Addressed
**a)** With what probability will there be one "second" among six randomly selected goblets?
**b)** With what probability will there be at least four "seconds" among twelve randomly selected goblets?
**c)** What is the expected number of "seconds" among 50 goblets? What is the associated variance?
#### Hint
The above questions can be addressed using the Binomial Distribution.
---
**Explanation:**
Using the Binomial Distribution is appropriate for these questions, as we are dealing with scenarios that have two possible outcomes (a goblet being either defective or not defective). Each event is independent, and we have a fixed number of trials (the number of goblets selected).
1. **Binomial Distribution Formula:**
- The Binomial probability $P(X = k)$ of getting exactly $k$ successes (defective goblets) out of $n$ trials is given by:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- $\binom{n}{k}$ is the binomial coefficient
- $p$ is the probability of success on an individual trial (here, $p = 0.1$)
- $n$ is the number of trials
2. **Expected Value and Variance:**
- The expected number of successes (defective goblets) in $n$ trials is $E(X) = np$.
- The variance of the number of successes is $Var(X) = np(1-p)$.
Ensure students understand these principles and encourage them to apply these formulas to solve the given problems.
---
This detailed explanation will aid students in comprehending how to use the Binomial Distribution to analyze and solve problems involving probabilities and expected values related to defective items in a production setting.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb19fb600-654f-4bb5-880f-15449a731741%2Fcc49a452-5411-440a-9074-468913dde639%2Fvd0pfia_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sure! Here is the transcription suitable for an educational website:
---
### Analysis of Defective Crystal Goblets Using Binomial Distribution
#### Problem Overview
A company producing fine crystal goblets has, in the past, experienced a defect rate of 10% in its goblets. Such goblets are designated as "seconds". Assume defective goblets are independent.
#### Questions to be Addressed
**a)** With what probability will there be one "second" among six randomly selected goblets?
**b)** With what probability will there be at least four "seconds" among twelve randomly selected goblets?
**c)** What is the expected number of "seconds" among 50 goblets? What is the associated variance?
#### Hint
The above questions can be addressed using the Binomial Distribution.
---
**Explanation:**
Using the Binomial Distribution is appropriate for these questions, as we are dealing with scenarios that have two possible outcomes (a goblet being either defective or not defective). Each event is independent, and we have a fixed number of trials (the number of goblets selected).
1. **Binomial Distribution Formula:**
- The Binomial probability $P(X = k)$ of getting exactly $k$ successes (defective goblets) out of $n$ trials is given by:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- $\binom{n}{k}$ is the binomial coefficient
- $p$ is the probability of success on an individual trial (here, $p = 0.1$)
- $n$ is the number of trials
2. **Expected Value and Variance:**
- The expected number of successes (defective goblets) in $n$ trials is $E(X) = np$.
- The variance of the number of successes is $Var(X) = np(1-p)$.
Ensure students understand these principles and encourage them to apply these formulas to solve the given problems.
---
This detailed explanation will aid students in comprehending how to use the Binomial Distribution to analyze and solve problems involving probabilities and expected values related to defective items in a production setting.
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