91Ó°ÊÓ

When working with scenarios where unusual events need quantification, probability calculations really shine. For this problem, the task is to find how likely it is that more than three cars will enter the mountain tunnel within a two-minute period. Knowing that, on average, one car enters every two minutes is our baseline statistic.
To solve this, we rely on the Poisson Distribution's probability mass function, which calculates the probability of a given number of events happening within a set period, given a known average rate (lambda, \( \lambda \)). In our example, \( \lambda = 1 \).
The Poisson probability formula is:
\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]
To find the probability for three events or fewer (and by extension, more than three), we calculate for \( k \) values of 0, 1, 2, and 3. Adding these probabilities gives us the cumulative probability, which we then use to find the complement for more than three events.

To find the probability of more than three cars entering in our two-minute window, we must first determine the cumulative probability of three or fewer cars entering. Cumulative probability accumulates the chances of all individual outcomes up to a certain point. It’s like building blocks, where each block (each probability) is stacked until we reach the desired height (cumulative probability).
We already found \( P(X = 0) = 0.3679 \), \( P(X = 1) = 0.3679 \), \( P(X = 2) = 0.1839 \), and \( P(X = 3) = 0.0613 \). Adding these probabilities together gives the cumulative probability:
\[ P(X \leq 3) = 0.3679 + 0.3679 + 0.1839 + 0.0613 = 0.981 \]
This means there's a 98.1% chance that three or fewer cars will enter the tunnel. Based on this, we use what is known as the complement rule:
\[ P(X > 3) = 1 - P(X \leq 3) = 1 - 0.981 = 0.019 \]
Therefore, there's a 1.9% probability of more than three cars entering, indicating it’s quite a rare event.

Statistical modeling helps us understand how likely certain events are by using mathematical frameworks. The Poisson Distribution is a powerful tool in statistical modeling, particularly for modeling "rare events" that happen independently over a given time period or space.
In this exercise, the problem perfectly fits the use of the Poisson distribution.

• You've got an average rate of occurrence (\( \lambda = 1 \))
• The events (cars entering the tunnel) are independent
• The interval of interest is fixed (two minutes)

With this statistical model, we can estimate the probability of observing different numbers of cars entering the tunnel. Using the Poisson distribution gives us a meaningful understanding of the practical implications and risks in this scenario, allowing us to prepare or adjust accordingly.
It's an advantage to use Poisson in such cases, as it’s simple yet provides a robust estimate of probabilities for events per interval, offering insights into how to handle situations safely.