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Chapter 5: Problem 58

On average a certain intersection results in 3 traffic accidents per month. What is the probability that for any given month at this intersection (a) exactly 5 accidents will occur? (b) less than 3 accidents will occur? (c) at least 2 accidents will occur?

Short Answer

Expert verified
The probability of: (a) exactly 5 accidents is found using the Poisson distribution formula for \( k=5 \), (b) less than 3 accidents is found by summing the Poisson probabilities for \( k=0,1,2 \) and (c) at least 2 accidents is found by subtracting the Poisson probabilities for \( k=0,1 \) from 1.

Step by step solution

01

Identify the parameters of the Poisson distribution

This given exercise problem can be modeled using a Poisson distribution with rate parameter \(\lambda\) equal to 3, which represents the average number of accidents per month.
02

Calculate the probability of exactly 5 accidents

The expression for a Poisson probability is: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where \( k \) is the number of occurrences, \( \lambda \) is the average rate and \( e \) is the natural number. Substituting \( k = 5 \) and \( \lambda = 3 \) into the expression, we find the probability for exactly 5 accidents.
03

Calculate the probability of less than 3 accidents

To calculate the probability of less than 3 accidents, we need to add up the probabilities of 0, 1, and 2 accidents. So, \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \). By substituting the values into the Poisson formula for each value of \( k \), we get the total probability.
04

Calculate the probability of at least 2 accidents

To calculate the probability of at least 2 accidents, we subtract the probabilities of 0 and 1 accidents from 1. This is given by the formula \( P(X \geq 2) = 1 - [P(X=0) + P(X=1)] \). By substituting the values in the Poisson formula for each value of k=0 and k=1, we get the total probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation is a crucial part of understanding events that may occur randomly, like traffic accidents at an intersection. In our scenario, the Poisson distribution is used to model such events. This distribution helps us compute the probability of a given number of accidents happening during a month.
  • The average number of accidents per month, represented by \( \lambda \), is 3.
  • The formula \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \) is used to find the probability of exactly \( k \) accidents.
  • For precisely 5 accidents: \( \lambda=3, k=5 \), substitute these values into the formula to get the probability.
  • For fewer than 3 accidents (0, 1, or 2), calculate probabilities for each and add them up.
  • For at least 2 accidents, compute the probability for 0 or 1 accident, subtract from 1.
By applying these steps, anyone can grasp how likely various accident outcomes are each month.
Traffic Accidents
Traffic accidents are regrettably common and can be studied using statistical models to improve safety measures. By assessing traffic patterns and probabilities, authorities can predict months with higher accident rates.In this problem, the setting focuses on an intersection prone to accidents. It experiences, on average, 3 accidents monthly. This average (\( \lambda = 3 \)) is essential for calculating how frequently other accident computations will occur.
  • Accidents can be heavily influenced by various factors like weather, vehicle conditions, or human behavior.
  • Analyzing accident data helps in resource allocation for emergency services and planning road improvements.
Understanding the nature of traffic accidents and modeling them statistically aids in advancing preventative measures and reactive strategies.
Statistical Modeling
Statistical modeling is the process of applying mathematical frameworks to understand or predict real-world phenomena. In this case, the Poisson distribution is an example of how statistical models help predict unlikely, random events like traffic accidents. The Poisson distribution is chosen here because it deals specifically with scenarios where the events occur independently, and the average number of occurrences is known.
  • Used when events happen at a constant rate over a fixed period of time.
  • Ideal for modeling rare events like traffic accidents on specific days or months.
  • Can assist in decision-making, such as evaluating whether to change traffic light timing or setup warning signs to reduce accident risk.
By understanding statistical modeling through Poisson distribution, not only can existing patterns be explained, but future probabilities can also be estimated efficiently.

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Most popular questions from this chapter

A production process produces electronic component parts. It has presumably been established that the probability of a defective part is \(0.01 .\) During a test of this presumption, 500 items are sampled randomly and 15 defective out of the 500 were observed. (a) What is your response to the presumption that the process is \(1 \%\) defective? Be sure that a computed probability accompanies your comment. (b) Under the presumption of a \(1 \%\) defective process, what is the probability that only 3 would be found defective? (c) Do (a) and (b) again using the Poisson approximation.

An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample: of 5 balls, find the probability that both blue balls and at. least 1 reel ball are selected.

An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the number of cars per year that will experience the fault is a Poisson random variable with \(\mathrm{A}=5\) (a) What is the probability that at most 3 cars per year will experience a catastrophe? (b) What is the probability that more than 1 car per year will experience a catastrophe?

Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one-half of its engines run, determine whether a 4 -engine plane or a 2 engine plane has the higher probability for a successful flight.

Suppose that for a very large shipment of integrated-circuit chips, the probability of failure for any one chip is \(0.10 .\) Assuming that the assumptions underlying the binomial distributions are met, find the probability that at most 3 chips fail in a random sample of 20.
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