Problem 5 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(1 \%\) of all items in a supermarket are not priced properly. A customer buys ten items. What is the probability that she will be delayed by the cashier because one or more of her items require a price check? Calculate both a binomial answer and a Poisson answer. Is the binomial model "exact" in this case? Explain.

Step-by-Step Solution

Answer

The probability of the customer being delayed due to an item needing a price check, as calculated using the binomial model, is 0.0956, and using the Poisson model, is 0.0952. The binomial model can be considered 'exact' in this case.

Step 1: Define the parameters for the binomial model

A binomial model is defined by two parameters: the number of trials (in this case, the number of items the customer buys, which is 10), and the probability of success on each trial (in this case, the probability that an item is not priced properly, which is 0.01).

Step 2: Calculate the binomial probability

To calculate the probability that one or more items require a price check using a binomial model, we need to find the complement of the probability that none of the items require a price check. The formula for a binomial probability is \(C(n, k) * p^k * (1 - p)^{n - k}\), where \(C(n, k)\) is the number of combinations of \(n\) items taken \(k\) at a time, \(p\) is the probability of success on each trial, and \(n - k\) is the number of failures. In this case, \(n = 10\), \( k = 0\), and \(p = 0.01\), so the probability that none of the items requires a price check is \(C(10, 0) * (0.01)^0 * (0.99)^{10 - 0} = 0.9044\). Thus, the probability that one or more items requires a price check is \(1 - 0.9044 = 0.0956\).

Step 3: Define the parameters for the Poisson model

A Poisson model is defined by a single parameter: the average number of successes in an interval. Here, the 'interval' is the customer's shopping trip, and a 'success' is buying an item that requires a price check. Given 10 items and a success rate of 0.01, the average number of successes is \(10 * 0.01 = 0.1\).

Step 4: Calculate the Poisson probability

Again, we need to find the complement of the probability that none of the items require a price check. The formula for a Poisson probability is \(e^{-\lambda} * (\lambda^k)/k!\), where \(e\) is Euler's number, \(\lambda\) is the average number of successes in an interval, \(k\) is the number of successes, and \(k!\) is the factorial of \(k\). In this case, \(\lambda = 0.1\) and \(k = 0\), so the probability that none of the items requires a price check is \(e^{-0.1} * (0.1^0)/0! = 0.9048\). Thus, the probability that one or more items require a price check is \(1 - 0.9048 = 0.0952\).

Step 5: Discuss whether the binomial model is exact

Here the binomial model can be considered 'exact' because we have a finite number of trials (10 items) and a constant probability of success (0.01) in each one. The Poisson model is an approximation that is used when dealing with very large numbers of trials, or when the probability of success is very small, which is not exactly the case here. Still, as you can see from our calculations, the Poisson model provides a probability very close to that of the binomial model, indicating it is a good approximation even in this case.

Key Concepts

Probability Theory

Probability theory is the mathematical framework that allows us to measure the likelihood of events occurring. It is the foundation for understanding lots of statistical concepts. In this particular scenario involving a customer buying items at a supermarket, probability theory is used to determine the chance that at least one of her items will need a price check.

This involves calculating the probability of an event not happening and subtracting it from 1 to get the probability of its complement - in this case, the item requiring a price check.

• **Sample Space**: All possible outcomes of an experiment. For our shopper, it's whether each item is priced correctly or not.
• **Event**: A subset of the sample space. For the cashier, the event of interest is items needing a price check.

With probability theory, we model uncertainties and predict the likelihood of events happening in diverse situations.

Poisson Distribution

The Poisson distribution is a statistical model used to estimate the probability of a number of events occurring within a fixed interval. It is especially useful for rare events where these occur independently.

For the supermarket example, the Poisson distribution is applied by determining the probability that one or more items need a price check out of the ten purchased. Despite having a relatively straightforward probability of occurrence (0.01 or 1%), it helps in providing an approximation.

• **Key Parameter**: The Poisson distribution relies on the average number of occurrences (\( \lambda \)) in a given time period or sample size. For the customer, this is given by multiplying the rate of mispricing (0.01) by her number of items (10).
• **Formula Usage**: Utilizing the Poisson probability formula \( e^{-\lambda} * \frac{\lambda^k}{k!} \) aids in computing the likelihood of observing a specified number of events (like 0 items needing a check).

The beauty of the Poisson distribution lies in its versatility and simplicity when dealing with events that have a small probability over a large number of trials.

Statistical Models

Statistical models are constructs that capture the patterns in data and describe the processes that generate the data. They help bridge real-world phenomena with theoretical probabilities. In the exercise of determining the likelihood of an item needing a price check, two statistical models were utilized: the binomial and Poisson models.

• **Binomial Model**: This model fits situations with a fixed number of trials (like our 10 items) where each trial has two possible outcomes - success (an item requires a price check) or failure (an item doesn’t require a price check). It accurately represents situations with limited trials and uniform probabilities.
• **Poisson Model**: Although used for approximation here, this model is typically best for situations with a large number of trials where events are rare. It becomes a useful tool to predict outcomes when direct calculations are complicated or unnecessary.

Both models provide frameworks to predict and interpret statistical data, aiding in informed decision-making processes.