A traffic intersection has an average of 10 cars passing through per minute. Find the probability that no cars pass through the intersection in a given minute

To solve this problem, we can use the Poisson distribution, which is suitable for modeling the number of events occurring within a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event.

1. Break Down the Problem

   We need to find the probability that no cars pass through the intersection in a given minute, given that the average number of cars passing through per minute is 10.

2. Relevant Concepts

   The Poisson probability mass function is given by: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ where:

   • $\lambda$ is the average rate (mean) of occurrence (10 cars per minute in this case),
   • $k$ is the number of occurrences we want to find the probability for (0 cars in this case),
   • $e$ is the base of the natural logarithm, approximately equal to 2.71828.

3. Analysis and Detail

   Substitute the given values into the Poisson formula: $P(X = 0) = \frac{10^0 \cdot e^{-10}}{0!}$ Simplifying this expression:

   • $10^0 = 1$
   • $0! = 1$
   • Thus, the expression becomes: $P(X = 0) = e^{-10}$

4. Verify and Summarize

   Calculate $e^{-10}$ using a calculator: $e^{-10} \approx 0.0000454$ Therefore, the probability that no cars pass through the intersection in a given minute is approximately 0.0000454.

Final Answer

The probability that no cars pass through the intersection in a given minute is approximately 0.0000454.