91Ó°ÊÓ

In this exercise, we are dealing with a Poisson distribution to calculate the probability of a specific event occurring. That event, in this case, is the number of accidents per year occurring at a particular intersection. The Poisson distribution is an excellent choice when dealing with rare or independent events that occur in a fixed interval of time or space. Formally, the probability mass function of a Poisson distribution is given by:\[ P(Y = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where:

• \(k\) is the number of occurrences we are interested in (e.g., 0, 1, 2, ..., 44 for finding \(P(Y < 45)\)).
• \(\lambda\) is the average rate of occurrence (36 accidents in our scenario).
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

To find the probability that at least 45 accidents happen, we look at the complementary probability, \(P(Y \geq 45)\). However, calculating it directly is complex, so we aim at finding \(P(Y < 45)\) and using the complement rule later.

The Cumulative Distribution Function (CDF) is a powerful tool in probability theory, especially for distributions like Poisson, where individual probability calculations can become tedious. The CDF for a Poisson distribution helps us find the probability that a random variable is less than or equal to a certain value, \(k\). It is the sum of probabilities of all outcomes from 0 to \(k\).Using the CDF for Poisson:\[ P(Y \leq 44) = \sum_{k=0}^{44} \frac{e^{-36} 36^k}{k!} \]This computation gives us the cumulative probability for all accident counts from 0 to 44. Instead of computing each probability manually, statistical software or a CDF calculator proves efficient and reduces the risk of arithmetic errors.The solution utilizes this CDF technique to simplify computing \(P(Y < 45)\). By incorporating all possible outcomes up to the desired threshold, the CDF provides an aggregated probability with much less computational effort. This step is crucial before applying the complement rule.

The complement rule in probability is a simple yet effective technique to find the probability of complex events. Specifically, it states that the probability of any event, \(A\), and its complement, \(A^c\), add up to 1:\[ P(A) = 1 - P(A^c) \]In the context of this problem, instead of calculating the probability of having 45 or more accidents directly, we determine it using its complement. The complementary event here is having fewer than 45 accidents, represented by \(P(Y < 45)\).Applying the complement rule:\[ P(Y \geq 45) = 1 - P(Y < 45) \]This approach greatly simplifies the calculation process. By finding \(P(Y \leq 44)\) using the Poisson CDF, which includes all probabilities from 0 to 44, we then subtract this cumulative probability from 1. As noted in the solution, if \(P(Y \leq 44)\) equals approximately 0.839, the resulting probability of needing an emergency redesign due to high accident rates becomes \(0.161\). This efficient method saves time and effort, providing clearer insights into probability outcomes.