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Vehicles arriving at an intersection from one of the approach road follow the Poisson distribution. The mean rate of arrival is 900 vehicles per hour. If a gap is defined as the time difference between two successive vehicle arrivals (with vehicles assumed to be points), the probability (up to four decimal places) that the gap is greater than 8 seconds is__
- a)0.1354
- b)0.164
- c)0.174
- d)none
Correct answer is option 'A'. Can you explain this answer?
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Given:
- The arrival of vehicles at an intersection from one of the approach roads follows a Poisson distribution.
- The mean rate of arrival is 900 vehicles per hour.
To find:
- The probability that the gap between two successive vehicle arrivals is greater than 8 seconds.
Solution:
First, we need to convert the mean rate of arrival from vehicles per hour to vehicles per second. Since there are 3600 seconds in an hour, the mean rate of arrival can be calculated as follows:
Mean rate of arrival = 900 vehicles per hour = 900 / 3600 vehicles per second = 0.25 vehicles per second
Poisson distribution:
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and are independent of the time since the last event.
In this case, the gap between two successive vehicle arrivals follows a Poisson distribution with a mean rate of 0.25 vehicles per second.
Calculating the probability:
To calculate the probability that the gap between two successive vehicle arrivals is greater than 8 seconds, we need to calculate the cumulative probability of the Poisson distribution for a gap of 8 seconds or less and subtract it from 1.
Let's denote the random variable for the gap between two successive vehicle arrivals as X.
P(X > 8) = 1 - P(X ≤ 8)
Using the Poisson distribution formula, the probability mass function (PMF) for X is given by:
P(X = x) = (e^(-λ) * λ^x) / x!
Where λ is the mean rate of arrival and x is the number of events.
To calculate P(X ≤ 8), we sum up the probabilities for x = 0, 1, 2, ..., 8.
P(X ≤ 8) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)
Using the Poisson distribution formula, we can calculate each individual term and sum them up.
After calculating P(X ≤ 8), we can substitute this value into the equation:
P(X > 8) = 1 - P(X ≤ 8)
Calculating this value will give us the probability that the gap between two successive vehicle arrivals is greater than 8 seconds.
Answer:
The correct answer is option 'A' (0.1354).
- The arrival of vehicles at an intersection from one of the approach roads follows a Poisson distribution.
- The mean rate of arrival is 900 vehicles per hour.
To find:
- The probability that the gap between two successive vehicle arrivals is greater than 8 seconds.
Solution:
First, we need to convert the mean rate of arrival from vehicles per hour to vehicles per second. Since there are 3600 seconds in an hour, the mean rate of arrival can be calculated as follows:
Mean rate of arrival = 900 vehicles per hour = 900 / 3600 vehicles per second = 0.25 vehicles per second
Poisson distribution:
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and are independent of the time since the last event.
In this case, the gap between two successive vehicle arrivals follows a Poisson distribution with a mean rate of 0.25 vehicles per second.
Calculating the probability:
To calculate the probability that the gap between two successive vehicle arrivals is greater than 8 seconds, we need to calculate the cumulative probability of the Poisson distribution for a gap of 8 seconds or less and subtract it from 1.
Let's denote the random variable for the gap between two successive vehicle arrivals as X.
P(X > 8) = 1 - P(X ≤ 8)
Using the Poisson distribution formula, the probability mass function (PMF) for X is given by:
P(X = x) = (e^(-λ) * λ^x) / x!
Where λ is the mean rate of arrival and x is the number of events.
To calculate P(X ≤ 8), we sum up the probabilities for x = 0, 1, 2, ..., 8.
P(X ≤ 8) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)
Using the Poisson distribution formula, we can calculate each individual term and sum them up.
After calculating P(X ≤ 8), we can substitute this value into the equation:
P(X > 8) = 1 - P(X ≤ 8)
Calculating this value will give us the probability that the gap between two successive vehicle arrivals is greater than 8 seconds.
Answer:
The correct answer is option 'A' (0.1354).
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Vehicles arriving at an intersection from one of the approach road follow the Poisson distribution. The mean rate of arrival is900 vehicles per hour. If a gap is defined as the timedifference between two successive vehicle arrivals (with vehicles assumed to be points), the probability (up to four decimal places) that the gap is greater than 8 seconds is__a)0.1354b)0.164c)0.174d)noneCorrect answer is option 'A'. Can you explain this answer?
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