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Chapter 5: Problem 76

A commuter airline receives an average of \(9.7\) complaints per day from its passengers. Using the Poisson formula, find the probability that on a certain day this airline will receive exactly 6 complaints.

Short Answer

Expert verified
The exact probability can be calculated using a calculator. Hence, up to Step 2, everything was done manually. The last step is usually done with the help of a calculator due to the complexity of computations involved.

Step by step solution

01

Understand the Poisson distribution

The Poisson distribution is given by the formula: \(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\) where \(\lambda\) is the average rate of occurrence, \(k\) is the exact number of occurrences we're interested in, and \(e\) is Euler's number (approximately equal to \(2.71828\)). In this problem, \(\lambda = 9.7\) and \(k = 6\).
02

Calculation Insertion to Formula

Plugging in these values into the Poisson formula we get: \(P(X = 6) = \frac{(9.7)^6 e^{-9.7}}{6!}\)
03

Calculate probability

Now, simply calculate the value of this expression to find the required probability. This will involve computing the factorials, the exponentials, and the multiplicative inverses.

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Key Concepts

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

Probability Calculation
When working with the Poisson distribution, a common task is to calculate the probability of a particular number of events occurring. This is determined using the Poisson probability formula:- The formula is: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]- \( P(X = k) \) represents the probability of \( k \) events occurring.- \( \lambda \) is the average rate of occurrence.To find the probability of exactly 6 complaints, we substitute \( \lambda = 9.7 \) and \( k = 6 \) into the equation.
  • Calculate \( 9.7^6 \): multiply 9.7 by itself six times.
  • Compute \( e^{-9.7} \): use the property of Euler's number to the negative power of the average rate.
  • Factor in the factorial of 6.
Once these operations are done, you'll derive the probability of this event occurring, which is the ultimate goal of applying the Poisson distribution formula in this context.
Average Rate of Occurrence
The Poisson distribution requires knowing the average rate of occurrence, denoted as \( \lambda \). This crucial parameter indicates how many events are expected during a fixed time frame, in this case, complaints per day.Understanding this concept:- **Measurement context:** In the exercise, \( \lambda = 9.7 \) reflects the average number of complaints received daily.- **Stability:** The average is assumed to remain consistent over time, aligning with the steady rate principle.The average rate \( \lambda \) is foundational for calculating probabilities and helps mathematically predict event frequencies in various scenarios.
Euler's Number
Euler's number, \( e \), is a fundamental constant used in many mathematical formulas, including the Poisson distribution. It is approximately equal to 2.71828 and comes into play prominently in calculus and exponential functions.In the context of the Poisson formula:- **Exponential decay:** It's used as \( e^{-\lambda} \), representing the exponential decay factor based on the average event rate.- **Calculation necessity:** Euler's number is crucial for accurately determining probabilities as it adjusts for the natural log scaling of time and occurrences.By including \( e \), you ensure that the probabilities reflect realistic patterns found in natural processes and event distributions.
Factorials in Poisson Distribution
Factorials are a key element in the Poisson formula, symbolized by \( k! \). A factorial, \[ n! \]is the product of all positive integers up to \( n \). For example, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).This component of the Poisson probability indicates the number of ways \( k \) events can occur, setting the scale for likelihood.- **Combinatorial nature:** Factorials account for permutations and combinations of events within the scope.- **Role in normalization:** In probability calculation, factorials help normalize the scale of probability, ensuring that different event sequences align correctly in probability.Understanding factorials is essential for applying and interpreting the Poisson distribution effectively.

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Most popular questions from this chapter

Twenty percent of the cars passing through a school zone are exceeding the speed limit by more than \(10 \mathrm{mph}\). a. Using the Poisson formula, find the probability that in a random sample of 100 cars passing through this school zone, exactly 25 will exceed the speed limit by more than \(10 \mathrm{mph}\). b. Using the Poisson probabilities table, find the probability that the number of cars exceeding the speed limit by more than 10 mph in a random sample of 100 cars passing through this school zone is \(\begin{array}{lll}\text { i. at most } 8 & \text { ii. } 15 \text { to } 20 & \text { iii. at least } 30\end{array}\)

A household receives an average of \(1.7\) pieces of junk mail per day. Find the probability that this household will receive exactly 3 pieces of junk mail on a certain day. Use the Poisson probability distribution formula.

Two teams, \(\mathrm{A}\) and \(\mathrm{B}\), will play a best-of-seven series, which will end as soon as one of the teams wins four games. Thus, the series may end in four, five, six, or seven games. Assume that each team has an equal chance of winning each game and that all games are independent of one another. Find the following probabilities. a. Team A wins the series in four games. b. Team A wins the series in five games. c. Seven games are required for a team to win the series

Let \(x\) be a Poisson random variable. Using the Poisson probabilities table, write the probability distribution of \(x\) for each of the following. Find the mean, variance, and standard deviation for each of these probability distributions. Draw a graph for each of these probability distributions. a. \(\lambda=1.3\) b. \(\lambda=2.1\)

Explain the meaning of the probability distribution of a discrete random variable. Give one example of such a probability distribution. What are the three ways to present the probability distribution of a discrete random variable?
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