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Chapter 5: Problem 62

A new automated production process averages 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day?

Short Answer

Expert verified
The probability of having three or more breakdowns during a day is approximately 0.191.

Step by step solution

01

Understanding the Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given the average number of times the event occurs over that interval. In this exercise, the average number of breakdowns is 1.5 per day.
02

Identify the Parameters

The parameter for the Poisson distribution is \( \lambda \), which represents the average rate. Here, \( \lambda = 1.5 \), which is the average number of breakdowns per day.
03

Poisson Probability Formula

The Poisson probability of observing \( k \) events is given by:\[P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\]Where \( k \) is the number of events, \( e \) is approximately 2.71828, and \( \lambda \) is 1.5.
04

Calculate Probability for 0, 1, and 2 Breakdowns

Using the formula from Step 3, calculate:\( P(X = 0) = \frac{e^{-1.5} \cdot 1.5^0}{0!} = e^{-1.5} \approx 0.22313 \)\( P(X = 1) = \frac{e^{-1.5} \cdot 1.5^1}{1!} = 1.5 \cdot e^{-1.5} \approx 0.33470 \)\( P(X = 2) = \frac{e^{-1.5} \cdot 1.5^2}{2!} = \frac{2.25 \cdot e^{-1.5}}{2} \approx 0.25103 \)
05

Calculate Probability of 3 or More Breakdowns

The probability of having three or more breakdowns is the complement of having 0, 1, or 2 breakdowns:\( P(X \geq 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) \)Substitute the values:\( P(X \geq 3) = 1 - (0.22313 + 0.33470 + 0.25103) \)\( P(X \geq 3) = 1 - 0.80886 \approx 0.19114 \)

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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 about figuring out the chance that a certain event will happen. In the context of the Poisson distribution, probabilities help us understand how likely it is that a certain number of events, like breakdowns, occurs in a given timeframe. In our exercise, we use the Poisson formula:
  • First, to calculate the probability of having zero breakdowns, one breakdown, or two breakdowns within a day.
  • The formula is: \(P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\), where \(\lambda\) is the average number of events.
  • By calculating these using the parameter \(\lambda = 1.5\), we can determine each probability for the values 0, 1, and 2.
This technique involves simple arithmetic and helps us understand how likely different scenarios are. By adding these probabilities and subtracting from one, we find the probability of three or more breakdowns.
Independent Events
In probability, when we say two events are independent, we're stating that the occurrence of one does not affect the other. This is an important concept in calculating the Poisson probability.
  • In our example of breakdowns, whether or not a machine breaks down in one time interval is independent of whether it breaks down in another.
  • This means the probability of having a breakdown today is just the same regardless of what happened yesterday or will happen tomorrow.
This independence helps justify the use of Poisson distribution because it assumes events happen independently over time. Without this independence, our calculations could become more complicated.
Random Events
Random events are those that happen unexpectedly and without a deterministic pattern. They can't be predicted exactly, but their probability can be modeled. In our example:
  • The machine breakdowns are considered random because they occur unpredictably during a day.
  • The Poisson distribution accommodates randomness by allowing us to calculate the probability of different numbers of events in fixed intervals, like days.
Understanding random events is crucial because it teaches us to view problems probabilistically, not deterministically. By modeling randomness, we are equipped with the tools to deal with uncertainty in real-world scenarios.
Statistical Modeling
Statistical modeling involves using mathematical frameworks to represent real-world processes. This helps in making predictions and understanding phenomena.
  • The Poisson distribution is a type of statistical model used to describe the likelihood of a given number of events happening in a fixed time or space.
  • It relies on knowing the average rate of events, \(\lambda\), which in our problem is the average breakdown rate per day.
  • By applying the Poisson model, we can predict probabilities concerning the production process, assisting in decision-making.
Statistical modeling thus transforms vague real-world problems into structured mathematical ones, allowing businesses to make informed, data-driven decisions. It bridges the gap between randomness and predictability, giving us a method to quantify uncertainty.

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Most popular questions from this chapter

Consider a Poisson distribution with \(\mu=3\) a. Write the appropriate Poisson probability function. b. \(\quad\) Compute \(f(2)\) c. \(\quad\) Compute \(f(1)\) d. Compute \(P(x \geq 2)\).

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