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Telephone calls come into an exchange according to a Poisson process with 5 calls per minute on the average. The probability that no call will come in during the two-minute period 10 A.M. to 10:02 A.M. on a particular day is
- a)0
- b)e-10
- c)2/5 e-5
- d)1 - e-10
Correct answer is option 'B'. Can you explain this answer?
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Telephone calls come into an exchange according to a Poisson process w...
λ = 5 x 2 = 10
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Telephone calls come into an exchange according to a Poisson process w...
To solve this problem, we can use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time when the events occur with a known constant mean rate and are independent of the time since the last event.
Let's break down the problem step by step:
Given:
- The average number of calls per minute is 5.
- We need to find the probability that no call will come in during the two-minute period from 10 A.M. to 10:02 A.M.
Step 1: Determine the average number of calls in a two-minute period.
Since the average number of calls per minute is 5, we can multiply it by 2 to get the average number of calls in a two-minute period:
Average number of calls in a two-minute period = 5 * 2 = 10
Step 2: Calculate the probability of no calls during the two-minute period.
The Poisson distribution formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
where P(x; λ) is the probability of x events occurring in the interval, λ is the average rate of events, e is the base of the natural logarithm, and x! is the factorial of x.
In our case, x = 0 (no calls) and λ = 10 (average number of calls in a two-minute period). Plugging these values into the formula, we get:
P(0; 10) = (e^(-10) * 10^0) / 0!
Step 3: Simplify the equation.
Anything raised to the power of 0 is 1, and 0! is equal to 1. Therefore, we can simplify the equation to:
P(0; 10) = e^(-10)
Step 4: Calculate the probability.
Using a calculator, we can find that e^(-10) is approximately 0.00004539992.
Therefore, the probability that no call will come in during the two-minute period from 10 A.M. to 10:02 A.M. is approximately 0.00004539992, which is approximately equal to e^(-10).
Hence, the correct answer is option 'B': e^(-10).
Let's break down the problem step by step:
Given:
- The average number of calls per minute is 5.
- We need to find the probability that no call will come in during the two-minute period from 10 A.M. to 10:02 A.M.
Step 1: Determine the average number of calls in a two-minute period.
Since the average number of calls per minute is 5, we can multiply it by 2 to get the average number of calls in a two-minute period:
Average number of calls in a two-minute period = 5 * 2 = 10
Step 2: Calculate the probability of no calls during the two-minute period.
The Poisson distribution formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
where P(x; λ) is the probability of x events occurring in the interval, λ is the average rate of events, e is the base of the natural logarithm, and x! is the factorial of x.
In our case, x = 0 (no calls) and λ = 10 (average number of calls in a two-minute period). Plugging these values into the formula, we get:
P(0; 10) = (e^(-10) * 10^0) / 0!
Step 3: Simplify the equation.
Anything raised to the power of 0 is 1, and 0! is equal to 1. Therefore, we can simplify the equation to:
P(0; 10) = e^(-10)
Step 4: Calculate the probability.
Using a calculator, we can find that e^(-10) is approximately 0.00004539992.
Therefore, the probability that no call will come in during the two-minute period from 10 A.M. to 10:02 A.M. is approximately 0.00004539992, which is approximately equal to e^(-10).
Hence, the correct answer is option 'B': e^(-10).
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Telephone calls come into an exchange according to a Poisson process with 5 calls per minute on the average. The probability that no call will come in during the two-minute period 10 A.M. to 10:02 A.M. on a particular day isa)0b)e-10c)2/5 e-5d)1 - e-10Correct answer is option 'B'. Can you explain this answer?
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Telephone calls come into an exchange according to a Poisson process with 5 calls per minute on the average. The probability that no call will come in during the two-minute period 10 A.M. to 10:02 A.M. on a particular day isa)0b)e-10c)2/5 e-5d)1 - e-10Correct answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Telephone calls come into an exchange according to a Poisson process with 5 calls per minute on the average. The probability that no call will come in during the two-minute period 10 A.M. to 10:02 A.M. on a particular day isa)0b)e-10c)2/5 e-5d)1 - e-10Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Telephone calls come into an exchange according to a Poisson process with 5 calls per minute on the average. The probability that no call will come in during the two-minute period 10 A.M. to 10:02 A.M. on a particular day isa)0b)e-10c)2/5 e-5d)1 - e-10Correct answer is option 'B'. Can you explain this answer?.
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