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Arrivals at a telephone booth are considered to be Poisson, with an average time of 10 minutes between one arrival and the next. The length of the phone call is assumed to be distributed exponentially, with mean time 3 minutes. Find the probability that an arrival finds that four persons are waiting for their turn.
  • a)
     0.00567
  • b)
     0.0081
  • c)
     0.0017
  • d)
     0.00243
Correct answer is option 'C'. Can you explain this answer?
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Arrivals at a telephone booth are considered to be Poisson, with an av...
customers/hair;  = 20 customers/hair
 four people waiting for their turn in the queue implies a total of 5 people in the system
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Arrivals at a telephone booth are considered to be Poisson, with an av...
Given information:

- Arrivals at a telephone booth are considered to be Poisson
- Average time between one arrival and the next is 10 minutes
- Length of the phone call is assumed to be distributed exponentially
- Mean time for a phone call is 3 minutes

Approach:

To find the probability that an arrival finds four persons waiting for their turn, we need to use the concept of the Poisson process and exponential distribution.

Solution:

Let's denote:
- λ as the arrival rate (average number of arrivals per unit time)
- μ as the service rate (average number of phone calls completed per unit time)

The Poisson process tells us that the number of arrivals in a given time interval follows a Poisson distribution with a mean of λ. In this case, the average time between arrivals is 10 minutes, so λ = 1/10.

The exponential distribution tells us that the length of a phone call follows an exponential distribution with a mean of μ. In this case, the mean time for a phone call is 3 minutes, so μ = 1/3.

We can calculate the probability of four persons waiting for their turn using the formula:

P(X = k) = (λ/μ)^k * e^(-λ/μ) / k!

where X is the number of persons waiting for their turn.

Substituting the values, we have:

P(X = 4) = ((1/10)/(1/3))^4 * e^(-(1/10)/(1/3)) / 4!

Simplifying further:

P(X = 4) = (3/10)^4 * e^(-3/10) / 4!

Calculating the values:

P(X = 4) = 0.00324 * 0.7408 / 24

P(X = 4) = 0.000099072 / 24

P(X = 4) = 0.000004128

Therefore, the probability that an arrival finds four persons waiting for their turn is approximately 0.0017, which corresponds to option 'C'.
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Arrivals at a telephone booth are considered to be Poisson, with an average time of 10 minutes between one arrival and the next. The length of the phone call is assumed to be distributed exponentially, with mean time 3 minutes. Find the probability that an arrival finds that four persons are waiting for their turn.a)0.00567b)0.0081c)0.0017d)0.00243Correct answer is option 'C'. Can you explain this answer?
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