Questions: On average, 3.5 traffic accidents per month occur at a certain intersection. Complete parts (a) through (c) below. Click here to view the table of Poisson probability sums. (a) What is the probability that exactly 7 accidents will occur in any given month at this intersection? The probability that exactly 7 accidents will occur in any given month at this intersection is (Round to four decimal places as needed.)

Solution Steps

To solve this problem, we will use the Poisson probability formula, which is suitable for calculating the probability of a given number of events happening in a fixed interval of time. The formula is:

\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]

where \( \lambda \) is the average number of events (3.5 in this case), \( k \) is the number of events we want the probability for (7 in this case), and \( e \) is the base of the natural logarithm.

We need to find the probability of exactly 7 traffic accidents occurring in a month at an intersection where the average number of accidents is 3.5 per month. This is a Poisson distribution problem.

The Poisson probability formula is:

\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]

where:

• \(\lambda = 3.5\)
• \(k = 7\)

Substitute the values into the formula:

\[ P(X = 7) = \frac{e^{-3.5} \cdot 3.5^7}{7!} \]

Calculate the expression:

\[ P(X = 7) \approx 0.0385 \]

Final Answer