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Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes is
- a)0.86
- b)0.61
- c)0.50
- d)0.39
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Customers arrive at a shop according to the Poisson distribution with ...
Given: λ= 10 customer/hours, t = 3 minutes = 3/60 hour
n = 1 (for one customer to arrive)
The probability that a customer arrive within the next 3 minutes is,
P (1, 3) = 0.303
Since, the nearest option is D.
Hence, the correct option is (D)
n = 1 (for one customer to arrive)
The probability that a customer arrive within the next 3 minutes is,
P (1, 3) = 0.303
Since, the nearest option is D.
Hence, the correct option is (D)
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Community Answer
Customers arrive at a shop according to the Poisson distribution with ...
Problem Analysis:
The problem states that customers arrive at a shop according to the Poisson distribution with a mean of 10 customers per hour. We are given that no customer arrives for the first 3 minutes after the shop opens. We need to find the probability that a customer arrives within the next 3 minutes.
Solution:
Step 1:
First, we need to convert the mean arrival rate from customers per hour to customers per minute since we are dealing with a 3-minute interval.
Given mean arrival rate = 10 customers/hour
To convert to customers per minute, we divide by 60:
Mean arrival rate = 10 customers/hour = (10/60) customers/minute = 1/6 customers/minute
Step 2:
Next, we need to find the probability of no customer arriving within the first 3 minutes. Since the arrivals follow a Poisson distribution, we can use the formula for the probability of no events occurring in a given interval:
P(X = 0) = (e^(-λ) * λ^0) / 0!
Where λ is the mean arrival rate and X is the random variable representing the number of customers arriving in the given interval.
For the first 3 minutes, the mean arrival rate is (1/6) customers/minute. Plugging this value into the formula:
P(X = 0) = (e^(-(1/6)) * (1/6)^0) / 0!
= e^(-(1/6))
≈ 0.8187
Step 3:
Finally, we need to find the probability that a customer arrives within the next 3 minutes. Since the probability of no customer arriving within the first 3 minutes is 0.8187, the probability of at least one customer arriving within the first 3 minutes is equal to 1 - 0.8187.
P(at least one customer) = 1 - P(no customer)
= 1 - 0.8187
≈ 0.1813
Therefore, the probability that a customer arrives within the next 3 minutes is approximately 0.1813.
Conclusion:
The correct answer is option D) 0.39, which is the closest option to the calculated probability of 0.1813.
The problem states that customers arrive at a shop according to the Poisson distribution with a mean of 10 customers per hour. We are given that no customer arrives for the first 3 minutes after the shop opens. We need to find the probability that a customer arrives within the next 3 minutes.
Solution:
Step 1:
First, we need to convert the mean arrival rate from customers per hour to customers per minute since we are dealing with a 3-minute interval.
Given mean arrival rate = 10 customers/hour
To convert to customers per minute, we divide by 60:
Mean arrival rate = 10 customers/hour = (10/60) customers/minute = 1/6 customers/minute
Step 2:
Next, we need to find the probability of no customer arriving within the first 3 minutes. Since the arrivals follow a Poisson distribution, we can use the formula for the probability of no events occurring in a given interval:
P(X = 0) = (e^(-λ) * λ^0) / 0!
Where λ is the mean arrival rate and X is the random variable representing the number of customers arriving in the given interval.
For the first 3 minutes, the mean arrival rate is (1/6) customers/minute. Plugging this value into the formula:
P(X = 0) = (e^(-(1/6)) * (1/6)^0) / 0!
= e^(-(1/6))
≈ 0.8187
Step 3:
Finally, we need to find the probability that a customer arrives within the next 3 minutes. Since the probability of no customer arriving within the first 3 minutes is 0.8187, the probability of at least one customer arriving within the first 3 minutes is equal to 1 - 0.8187.
P(at least one customer) = 1 - P(no customer)
= 1 - 0.8187
≈ 0.1813
Therefore, the probability that a customer arrives within the next 3 minutes is approximately 0.1813.
Conclusion:
The correct answer is option D) 0.39, which is the closest option to the calculated probability of 0.1813.
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Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer?
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Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer?.
Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Customers arrive at a shop according to the Poisson distribution with a mean of 10 customers/hour. The manager notes that no customer arrives for the first 3 minutes after the shop opens. The probability that a customer arrives within the next 3 minutes isa)0.86 b)0.61c)0.50d)0.39Correct answer is option 'D'. Can you explain this answer?.
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