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Chapter 5: Problem 98

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 \(10 \mathrm{mph}\) in a random sample of 100 cars passing through this school zone is i. at most \(\underline{8}\) ii. 15 to 20 iii. at least 30

Short Answer

Expert verified
The probabilities can be found by calculating or by using a table, as shown in the solution steps. However, the actual numerical answer requires calculating with given values.

Step by step solution

01

Calculation for exactly 25 cars

In a random sample of 100 cars, the expected number of cars \(\mu\) exceeding the speed limit is \(0.20 \times 100 = 20\). Since we are looking for the probability that exactly 25 cars are speeding, we should substitute \( x = 25 \) and \( \mu = 20 \) into the Poisson formula: \[ P(25; 20) = \frac{e^{-20} \cdot 20^{25}}{25!} \]
02

Calculation for at most 8 cars

To find the probability that at most 8 cars are exceeding the speed limit, we sum up the probabilities that exactly 0, 1, 2,..., 8 cars are speeding. These probabilities can be obtained from the Poisson probabilities table or calculated by substituting \( x = 0, 1, 2,..., 8 \) into the Poisson formula, each time with \( \mu = 20 \). \[ P(X \leq 8) = P(0; 20) + P(1; 20) + P(2; 20) + ... + P(8; 20) \]
03

Calculation for 15 to 20 cars

The probability that the number of speeding cars is between 15 and 20 (inclusive) is obtained by summing the probabilities for each of these outcomes: exactly 15, 16,鈥�, or 20 cars. Again, use the Poisson probabilities table or compute these probabilities by inputting \( x = 15, 16,..., 20 \) into the Poisson formula, each time with \( \mu = 20 \). \[ P(15 \leq X \leq 20) = P(15; 20) + P(16; 20) + ... + P(20; 20) \]
04

Calculation for at least 30 cars

The probability that at least 30 cars are speeding can be found using the complement rule of probability: the probability of an event happening is 1 minus the probability that it doesn't happen. Therefore, subtract the sum of probabilities that 0, 1, 2,..., 29 cars are speeding from 1. These probabilities can be obtained from the Poisson probabilities table or computed by substituting \( x = 0, 1, 2,..., 29 \) into the Poisson formula, each time with \( \mu = 20 \). \[ P(X \geq 30) = 1 - [P(0; 20) + P(1; 20) + ... + P(29; 20)] \]

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Key Concepts

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

Probability Calculations
The Poisson distribution is a powerful tool for probability calculations, especially when dealing with events over a fixed interval or in a defined space. It helps us determine the likelihood of a given number of events occurring. In this exercise, the focus is on cars exceeding the speed limit in a school zone.
Given that 20% of each passing car exceeds the speed limit, our first probability calculation involves finding the chance that exactly 25 out of 100 cars speed. Here, the expected number of speeding cars (\( \mu \)) is 20. Using the Poisson formula:
  • Formula: \( P(x; \mu) = \frac{e^{-\mu} \cdot \mu^x}{x!} \)
  • Example Calculation: For exactly 25 cars, plug in \( x = 25 \) and \( \mu = 20 \) to find \( P(25; 20) \).
Simplified use of the Poisson formula allows us to calculate these probabilities efficiently.
Complement Rule
The complement rule in probability is a useful concept to simplify calculations for events expressed as 'at least' scenarios. In this context, we seek the probability that at least 30 cars exceed the speed limit.
The complement rule says that the probability of an event happening is 1 minus the probability of the event not happening. Thus, instead of finding probabilities directly for 30 or more cars speeding, we find the complement:
  • : Calculate the probability for 0 to 29 cars speeding, sum them, and subtract from 1.
  • : Formula: \( P(X \geq 30) = 1 - [P(0; 20) + P(1; 20) + ... + P(29; 20)] \)
The complement rule is a simple yet effective method to tackle probabilities quickly, especially when dealing with large intervals.
Random Sample
In probability and statistics, a random sample represents a subset of items drawn from a larger population where each item has an equal probability of being chosen. This ensures that the sample is representative of the entire population.
In our exercise, the random sample consists of 100 cars passing through the school zone.
Important attributes of a random sample include:
  • : Each car has the same chance of selection.
  • : Statistics derived from the sample are used to infer about the population.
Considering random samples is significant in probability calculations because it validates the use of statistical distributions like the Poisson distribution for predicting event likelihoods.
Expected Value
The expected value, often denoted as \( \mu \), is a critical concept when dealing with probability distributions. It provides a measure of the center or 'average' outcome where the distribution is centered.
For Poisson distributions, the expected value is the product of the rate (percentage of cars speeding) and the sample size.
Utilizing the expected value allows the Poisson formula to effectively calculate probabilities:
  • : Example in Context: 20% of 100 cars indicates \( \mu = 20 \).
  • : A measure of central tendency in the Poisson distribution.
This concept helps in grounding our probability calculations, providing a reference point for comparison and prediction.

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Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(x=0)\) for \(n=5\) and \(p=.05\) b. \(P(x=4)\) for \(n=7\) and \(p=.90\) c. \(P(x=7)\) for \(n=10\) and \(p=.60\) Verify your answers by using Table I of Appendix \(\mathrm{C}\).
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