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Magazine Chapter 3: Problem 128
Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?
Short Answer
Expert verified
The probability is approximately 0.7364.
Step by step solution
01
Understand the Poisson process
The Poisson process describes the number of events happening in a fixed interval of time. The mean rate given here is 80 cars per hour, which means on average, one car arrives every 0.75 minutes (since there are 60/80 = 0.75 minutes per car).
02
Convert the mean rate to the desired time interval
Since the phone call lasts one minute, we need to find the mean number of cars arriving in this one-minute interval. Given that 80 cars arrive per 60 minutes, the mean number of cars arriving in one minute is \(\lambda = \frac{80}{60} = \frac{4}{3}\) cars per minute.
03
Use the Poisson probability formula
The probability that exactly k events occur in a Poisson process is given by \(P(k;\lambda) = \frac{{e^{-\lambda} \lambda^k}}{{k!}}\). Where \(\lambda = \frac{4}{3}\) and k is the number of cars arriving.
04
Find the probability of no cars arriving
Calculate the probability of 0 cars arriving during the one-minute call using \(P(0; \lambda) = \frac{{e^{-(4/3)} (4/3)^0}}{{0!}} = e^{-(4/3)}\).
05
Calculate the probability that at least 1 car arrives
The probability that at least one car arrives is the complement of the probability that no cars arrive: \(P(\geq 1) = 1 - P(0; \lambda) = 1 - e^{-(4/3)}\).
06
Compute the numerical value
Calculate the numerical value using the exponential function: \(P(\geq 1) = 1 - e^{-(4/3)} \approx 1 - 0.2636 = 0.7364\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Theory
Probability theory is the mathematical framework for quantifying uncertainty. It helps us understand the likelihood of different outcomes in uncertain situations.
In the context of the Poisson process, probability theory is used to model how events, like cars arriving at a toll booth, are distributed over time.
In the context of the Poisson process, probability theory is used to model how events, like cars arriving at a toll booth, are distributed over time.
- The mean rate, here given as 80 cars per hour, represents the average number of times an event is expected to occur in a fixed period.
- Probability models, such as the Poisson distribution, rely on this mean rate to calculate how probable certain numbers of events are within a specific time frame.
- This helps in understanding the probability of events happening in short intervals, such as during the one-minute phone call in this problem.
Event Arrival
Event arrival in a Poisson process refers to how and when events occur over a period of time.
The Poisson process is particularly applicable when the events occur randomly and independently of each other.
The Poisson process is particularly applicable when the events occur randomly and independently of each other.
- In this example, events are the arrival of cars, and they are assumed to follow a specific, predictable statistical behavior.
- The Poisson process provides a framework to calculate the probability of a certain number of cars arriving at the toll both during the phone call.
- Understanding event arrival is crucial to properly applying the Poisson probability formula.
Exponential Function
The exponential function plays a key role in the calculations of a Poisson process.
It helps determine the probability of no events happening, which can then be used to find the probability of one or more events.
It helps determine the probability of no events happening, which can then be used to find the probability of one or more events.
- The formula involved is the Poisson probability formula: \(P(k; \lambda) = \frac{{e^{-\lambda} \lambda^k}}{{k!}}\), where \(e\) denotes the exponential function.
- The exponential function \(e^{-\lambda}\) calculates the probability of zero cars arriving in this scenario.
- It is derived because as \(\lambda\) increases, the probability of no events decreases. This exponential decay explains why we find it useful in probability distribution calculations.
- It is derived because as \(\lambda\) increases, the probability of no events decreases. This exponential decay explains why we find it useful in probability distribution calculations.
Complement Rule
The complement rule is a basic but powerful concept in probability theory.
It is used to find the probability of an event happening by subtracting the probability that the event does not happen from 1.
It is used to find the probability of an event happening by subtracting the probability that the event does not happen from 1.
- In our problem, the event of interest is at least one car arriving during the call. Thus, the complement event is no cars arriving.
- We first compute the probability for the complement (no cars) using the Poisson probability formula \(P(0; \lambda)\).
- Then, applying the complement rule yields \(P(\geq 1) = 1 - P(0; \lambda)\).
