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On an average a box contains 2 defective items. Find the probability that a randomly selected box has
a) no defective item
b) atleast 2 defective items.?
a) no defective item
b) atleast 2 defective items.?
Most Upvoted Answer
On an average a box contains 2 defective items. Find the probability t...
Certainly! Let's delve into the probability of finding defective items in a box.
Understanding the Problem
On average, a box contains 2 defective items. This scenario can be modeled using the Poisson distribution, which is suitable for counting the number of events (defective items) in a fixed interval (a box) when these events occur with a known constant mean rate.
Parameters of the Poisson Distribution
- Mean (λ) = 2 defective items
- The probability mass function for the Poisson distribution is given by:
\[
P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}
\]
where:
- \(X\) = number of defective items
- \(k\) = specific number of defective items
- \(e\) ≈ 2.71828
Calculating the Probabilities
1. Probability of No Defective Items (k=0)
\[
P(X = 0) = \frac{e^{-2} \cdot 2^0}{0!} = e^{-2} \approx 0.1353
\]
2. Probability of At Least 2 Defective Items
To find the probability of at least 2 defective items, we calculate:
\[
P(X \geq 2) = 1 - P(X < 2)="1" -="" (p(x="0)" +="" p(x="" />
\]
- Probability of 1 Defective Item (k=1)
\[
P(X = 1) = \frac{e^{-2} \cdot 2^1}{1!} = 2e^{-2} \approx 0.2707
\]
- Combining the Probabilities
\[
P(X < 2)="P(X" =="" 0)="" +="" p(x="1)" \approx="" 0.1353="" +="" 0.2707="" />
\]
- Finally,
\[
P(X \geq 2) = 1 - 0.4060 \approx 0.5940
\]
Summary of Results
- Probability of No Defective Items: Approximately 0.1353
- Probability of At Least 2 Defective Items: Approximately 0.5940
This analysis demonstrates how to effectively use the Poisson distribution to assess the likelihood of defective items in a box.
Understanding the Problem
On average, a box contains 2 defective items. This scenario can be modeled using the Poisson distribution, which is suitable for counting the number of events (defective items) in a fixed interval (a box) when these events occur with a known constant mean rate.
Parameters of the Poisson Distribution
- Mean (λ) = 2 defective items
- The probability mass function for the Poisson distribution is given by:
\[
P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}
\]
where:
- \(X\) = number of defective items
- \(k\) = specific number of defective items
- \(e\) ≈ 2.71828
Calculating the Probabilities
1. Probability of No Defective Items (k=0)
\[
P(X = 0) = \frac{e^{-2} \cdot 2^0}{0!} = e^{-2} \approx 0.1353
\]
2. Probability of At Least 2 Defective Items
To find the probability of at least 2 defective items, we calculate:
\[
P(X \geq 2) = 1 - P(X < 2)="1" -="" (p(x="0)" +="" p(x="" />
\]
- Probability of 1 Defective Item (k=1)
\[
P(X = 1) = \frac{e^{-2} \cdot 2^1}{1!} = 2e^{-2} \approx 0.2707
\]
- Combining the Probabilities
\[
P(X < 2)="P(X" =="" 0)="" +="" p(x="1)" \approx="" 0.1353="" +="" 0.2707="" />
\]
- Finally,
\[
P(X \geq 2) = 1 - 0.4060 \approx 0.5940
\]
Summary of Results
- Probability of No Defective Items: Approximately 0.1353
- Probability of At Least 2 Defective Items: Approximately 0.5940
This analysis demonstrates how to effectively use the Poisson distribution to assess the likelihood of defective items in a box.
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On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.?
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On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.?.
On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On an average a box contains 2 defective items. Find the probability that a randomly selected box hasa) no defective itemb) atleast 2 defective items.?.
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