The number of accidents (𝐴) at an intersection is counted for a 12-hour period. The number of accidents for a 12-hour period follows a Poisson distribution with a parameter value of 2.Calculate P(A=5)
Question
The number of accidents (𝐴) at an intersection is counted for a 12-hour period. The number of accidents for a 12-hour period follows a Poisson distribution with a parameter value of 2.Calculate P(A=5)
Solution
The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. The formula for the Poisson distribution is:
P(X=k) = λ^k * e^-λ / k!
where:
- P(X=k) is the probability of k events,
- λ is the average rate of value,
- e is the base of the natural logarithm (approximately equal to 2.71828),
- k! is the factorial of k.
In this case, we are asked to find the probability of exactly 5 accidents (k=5) in a 12-hour period, given that the average rate of accidents is 2 per 12-hour period (λ=2).
Substituting these values into the formula, we get:
P(A=5) = 2^5 * e^-2 / 5!
Calculating this gives:
P(A=5) = 32 * e^-2 / 120
Now, we just need to calculate the value of e^-2 and then multiply and divide as necessary. The value of e^-2 is approximately 0.13534.
So, P(A=5) = 32 * 0.13534 / 120 = 0.036089
So, the probability of exactly 5 accidents occurring in a 12-hour period is approximately 0.036 or 3.6%.
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