Problem 20 - Applied Probability

Question

One car in 5 in a certain community is a Volvo. If the number of cars passing a traffic check point in an hour is Poisson (130), what is the expected number of Volvos? What is the probability of at least 30 Volvos? What is the probability the number of Volvos is between 16 and 40 (inclusive)?

Step-by-Step Solution

Answer

The expected number of Volvos is 26. The probability of observing at least 30 Volvos and for observing between 16 and 40 Volvos need to be calculated using a Poisson distribution with the given average.

Step 1: Determine the average number of Volvos

First, calculate the expected number of Volvos by multiplying the total average number of cars by the probability that any given car is a Volvo. Since one car in every five is a Volvo, the probability is 0.2 (or 1/5). If the average number of cars passing is 130 per hour, the average number of Volvos would be 130 multiplied by 0.2.

Step 2: Calculate the probability of at least 30 Volvos

To calculate the probability of at least 30 Volvos, use the cumulative distribution function (CDF) of the Poisson distribution for P(X >= 30). However, most tables provide P(X <= x), so you would calculate P(X < 30) and subtract it from 1 to get P(X >= 30).

Step 3: Calculate the probability of 16 to 40 Volvos

To find the probability that the number of Volvos is between 16 and 40, inclusive, calculate P(X <= 40) - P(X < 16). This can be done using the CDF for cumulative probabilities of a Poisson distribution. Since the CDF gives P(X <= x), P(X < 16) is actually P(X <= 15).

Key Concepts

Cumulative Distribution Function

Understanding the cumulative distribution function (CDF) is pivotal when dealing with probability distributions. In the context of probability theory, the CDF represents the probability that a random variable takes a value less than or equal to a specific value. It essentially tells us the cumulative probability of an outcome and is a fundamental tool for calculating probabilities like 'at least' or 'between' certain values.

For instance, let's consider the exercise with the community's cars. To find the probability of observing at least 30 Volvos passing the checkpoint, we would typically calculate the CDF for all values up to 29 and subtract this from 1. This technique provides the cumulative probability of observing 30 or more Volvos. Similarly, to find the probability of the number of Volvos being between 16 and 40, which requires calculating the probability for the range, we use the CDF values at the endpoints of the range. Here, you'd compute the CDF for 40 and subtract the CDF for 15, giving you the probability for the number being between 16 and 40.

An improved explanation ensures students understand that the CDF is not just a formula to plug values into, but a concept that sums probabilities up to a given threshold, making it a cornerstone for understanding many types of probability scenarios.

Expected Value

The expected value, often represented as the mean of a probability distribution, is a crucial concept in statistics and probability theory. It provides a measure of the 'center' of the distribution and can be thought of as the long-term average if an experiment is repeated many times.

In our traffic checkpoint scenario, the expected value offers insight into the number of Volvos we predict to pass within an hour, based on the proportion of Volvos in the community. To calculate the expected value, multiply the total number of cars by the likelihood of any single car being a Volvo: 130 cars per hour times the probability of 0.2 results in an expected value of 26 Volvos per hour.

Understanding the expected value is critical as it forms the basis for more complex interpretations in statistics, helps in decision making, and is frequently used in realms like insurance, finance, and economics. It is a foundational pillar in understanding the behavior of random variables over time.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena. It provides the framework and tools to quantify uncertainty and make predictions about outcomes. This theory is the bedrock upon which statistical inference and many aspects of machine learning are constructed.

Within the context of Poisson distribution, as in our exercise involving Volvo cars, probability theory helps us model and predict the number of events occurring within a fixed interval, given a known average rate of occurrence. Using probability theory, we can calculate the likelihood of observing a certain number of cars (Volvos, in our case) passing a checkpoint, which is instrumental for city planning, traffic management, and resource allocation.

The application of probability theory in solving practical problems, like the textbook exercise, is key to its understanding. It's not just the calculation but also the interpretation of results that provides valuable insights, thereby emphasizing the importance of mastering this theory for a deeper grasp of how random processes work in the real world.