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At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?
Correct answer is '0.0173'. Can you explain this answer?
Verified Answer
At a production machine, parts arrive according to a Poisson process ...
Given that
λ = 0.35 per minute
µ = 1 /2 = 0.5 per minute
Hence,
ρ = λ/ µ = 0.7
The probability that a random part arrival finds that there are already 8 parts in the system is Pn = ρn (1−ρ) = 0.78 (1−0.7) = 0.01729
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At a production machine, parts arrive according to a Poisson process ...
Probability of Finding 8 Parts in the System
To find the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution and exponential distribution.

Poisson Process
A Poisson process is a mathematical model used to describe the arrival of events over a continuous interval of time. In this case, the parts arriving at the production machine follow a Poisson process with a rate of 0.35 parts per minute.

Exponential Distribution
The processing time for each part follows an exponential distribution with a mean of 2 minutes. The exponential distribution is commonly used to model the time between events in a Poisson process.

Probability Calculation
To calculate the probability that a random part arrival finds 8 parts in the system, we can use the Poisson distribution. The Poisson distribution gives the probability of a certain number of events occurring in a fixed interval of time, given the average rate of events.

The formula for the Poisson distribution is:

P(X = k) = (e^(-λ) * λ^k) / k!

Where:
- P(X = k) is the probability of k events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average rate of events (in this case, 0.35 parts per minute)
- k is the number of events

In this case, we want to find the probability of finding 8 parts in the system, so k = 8. We can substitute these values into the formula:

P(X = 8) = (e^(-0.35) * 0.35^8) / 8!

Calculating this expression gives us the probability of finding 8 parts in the system. The correct answer is 0.0173 or approximately 1.73%.

Therefore, the probability that a random part arrival finds 8 parts in the system is 0.0173.
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At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts have exponential distribution with mean of 2 min. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in queue)?Correct answer is '0.0173'. Can you explain this answer?
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