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Chapter 17: Problem 6

Passengers arrive at a security checkpoint in a busy airport at the rate of 8 per 10 -minute period. For the time between 8: 00 and 8: 10 on a specific day, find the probability that a) 8 passengers arrive b) no more than 5 passengers arrive c) at least 4 passengers arrive.

Short Answer

Expert verified
a) 0.1425; b) 0.1912; c) 0.9473

Step by step solution

01

Identify the Distribution

In this exercise, we are dealing with a Poisson distribution since we are given a number of arrivals over a fixed period of time. The average rate (\( \lambda \)) of arrivals is given as 8 passengers per 10-minute period.
02

Formulate the Probability for 8 Arrivals

To find the probability of exactly 8 passengers arriving between 8:00 and 8:10, we use the Poisson probability formula: \[ P(X=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]Where \( \lambda = 8 \) and \( k = 8 \). Calculate:\[ P(X=8) = \frac{e^{-8} \cdot 8^8}{8!} \]
03

Probability Calculation for No More Than 5 Arrivals

To find the probability that no more than 5 passengers arrive, we need to calculate the cumulative probability for \( k = 0 \) to \( k = 5 \). Sum these individual probabilities using the formula from Step 2. \[ P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-8} \cdot 8^k}{k!} \]
04

Probability Calculation for At Least 4 Arrivals

To find the probability that at least 4 passengers arrive, calculate the cumulative probability from 0 to 3 and subtract it from 1.\[ P(X \geq 4) = 1 - P(X < 4) = 1 - \sum_{k=0}^{3} \frac{e^{-8} \cdot 8^k}{k!} \]
05

Perform Calculations

Using a calculator or statistical software, compute the values from Steps 2-4:- For part (a), calculate \( P(X=8) \).- For part (b), sum \( P(X=0) \) to \( P(X=5) \) to find \( P(X \leq 5) \).- For part (c), calculate \( P(X \geq 4) = 1 - P(X \leq 3) \).

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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 a branch of mathematics that deals with the analysis of random phenomena. The outcomes of random events can often be modeled using probability distributions.
These distributions provide us with a framework to make predictions about uncertain events.

The Poisson distribution is specifically used in scenarios that involve counting the number of events in a fixed interval of time or space.
  • It requires two key parameters: the average number of events in the interval (denoted by \( \lambda \)) and the actual number of occurrences we are interested in (denoted by \( k \)).
  • The formula for the Poisson probability of observing \( k \) events is \( P(X=k) = \frac{e^{-{\lambda}} \cdot \lambda^k}{k!} \).
  • The function \( e \) represents Euler's number, approximately 2.718. It is the base of the natural logarithm and plays a vital role in continuous processes.
By understanding these parameters and how the formula works, students can better interpret various problems relating to arrival rates, queues, and counts of events in certain intervals.
Arrival Rate
The arrival rate in the context of Poisson distribution indicates the average number of events or arrivals within a specified time period.
This notion is crucial when examining situations like those at a security checkpoint, where arrivals are random yet average over time.

In the initial exercise, the arrival rate is given as 8 passengers per 10-minute period.
  • This is represented by the parameter \( \lambda \) within the Poisson distribution formula.
  • By knowing \( \lambda \), we can compute the probability for different numbers of passenger arrivals.
  • The rate is crucial in setting the expectation for a process over time and lays the groundwork for further calculations using Poisson's formula.
Grasping this concept allows students to apply these ideas to other problems involving consistent arrival processes across different sectors.
Cumulative Probability
Cumulative probability refers to the probability that a random variable takes a value less than or equal to, or greater than or equal to, a certain number.
It incorporates all individual probabilities up to a certain point.

In solving Poisson distribution problems, cumulative probabilities are often calculated to answer queries about ranges of occurrences.
  • To find the probability of no more than 5 passengers arriving, we can sum the individual probabilities for 0 to 5 arrivals.
  • The formula is \( P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-8} \cdot 8^k}{k!} \).
  • For cases where at least a certain number of events are required, such as at least 4 arrivals, we need to subtract from 1 the probability of fewer events: \( P(X \geq 4) = 1 - \sum_{k=0}^{3} \frac{e^{-8} \cdot 8^k}{k!} \).
This technique of summing probabilities is beneficial to transition from discrete probabilities to encompassing broader ranges in predictions and offers insight into the likelihood of more complex outcomes.

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Most popular questions from this chapter

The time to first failure of an engine valve (in thousands of hours) is a random variable Y with a pdf \(f(y)=\left\\{\begin{array}{ll}2 y e^{-y^{2}} & 0 \leqslant y \\ 0 & \text { otherwise }\end{array}\\.\) a) Show that \(f(y)\) satisfies the requirements of a density function. b) Find the probability that the valve will last at least 2000 hours before being serviced. c) Find the mean of the random variable. (Hint: You need to know that \(\lim _{x \rightarrow \infty} \int_{0}^{x} e^{-t^{2}} d t=\frac{\sqrt{\pi}}{2}\) ) d) Find the median of \(Y\) e) Find the IQR. f) An engine utilizes two such valves and needs servicing as soon as any of the two valves fail. Find the probability that the engine needs servicing before 200 hours of work.

Some flashlights use one AA-type battery. The voltage in any new battery is considered acceptable if it is at least 1.3 volts. \(90 \%\) of the AA batteries from a specific supplier have an acceptable voltage. Batteries are usually tested till an acceptable one is found. Then it is installed in the flashlight. Let \(X\) be the number of batteries that must be tested. a) What is \(P(1),\) i.e. \(P(x=1) ?\) b) What is \(P(2) ?\) c) What is \(P(3) ?\) d) To have \(x=5,\) what must be true of the fourth battery tested? of the fifth one? e) Use your observations above to obtain a general model for \(P(x)\).

A soft-drink machine can be regulated so that it discharges an average \(\mu\) cc per bottle. If the amount of fill is normally distributed with a standard deviation \(9 \mathrm{cc}\) give the setting for \(\mu\) so that \(237 \mathrm{cc}\) bottles will overflow only \(1 \%\) of the time.

A machine produces bearings with diameters that are normally distributed with mean \(3.0005 \mathrm{cm}\) and standard deviation \(0.0010 \mathrm{cm} .\) Specifications require the bearing diameters to lie in the interval \(3.000 \pm 0.0020 \mathrm{cm} .\) Those outside the interval are considered scrap and must be disposed of. What fraction of the production will be scrap?

Houses in a large city are equipped with alarm systems to protect them from burglary. A company claims their system to be \(98 \%\) reliable. That is, it will trigger an alarm in \(98 \%\) of the cases. In a certain neighbourhood, 10 houses equipped with this system experience an attempted burglary. a) Find the probability that all the alarms work properly. b) Find the probability that at least half of the houses trigger an alarm. c) Find the probability that at most 8 alarms will work properly.
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