Question

A company that manufactures video cameras produces a basic model and a deluxe model. Over the past year, \(40 \%\) of the cameras sold have been the basic model. Of those buying the basic model, \(30 \%\) purchase an extended warranty, whereas \(50 \%\) of all purchasers of the deluxe model buy an extended warranty. If you learn that a randomly selected purchaser bought an extended warranty, what is the probability that he or she has a basic model?

Short answer

The probability that a randomly selected purchaser bought a basic model given that they bought an extended warranty is approximately \(0.2857\) or \(28.57\% \).

Step-by-step solution

Understanding the Problem Statement

Let's denote \(B\) for the event 'purchaser bought a basic model' and \(W\) for the event 'purchaser bought an extended warranty'. We know from the problem that \(P(B) = 0.4\) (probability that a purchaser bought a basic model), \(P(W|B) = 0.3\) (probability that a purchaser bought an extended warranty given a basic model was bought), \(P(W|B') = 0.5\) (probability that a purchaser bought an extended warranty given a deluxe model was bought). The task is to find \(P(B|W)\) (probability that a purchaser bought a basic model given an extended warranty was bought).

Use of Total Probability Theorem

First, we need to find \(P(W)\), the probability that a purchaser bought an extended warranty. We use total probability theorem here, \(P(W) = P(W and B) + P(W and B') = P(W|B)P(B) + P(W|B')P(B') = 0.3 * 0.4 + 0.5 * 0.6 = 0.12 + 0.3 = 0.42.

Use of Bayes' Theorem

Next, we use Bayes' theorem to calculate the conditional probability \(P(B|W)\). The formula for Bayes' theorem is \(P(B|W) = {P(W|B)P(B)}/{P(W)} = {0.3*0.4}/{0.42} ~= 0.2857.

Key concepts

Total Probability Theorem

Understanding the Total Probability Theorem (TPT) is foundational for solving many problems in statistics, especially when dealing with complex probability models that encompass multiple events. Let's think of TPT as a tool that helps us to weigh and combine the likelihood of several distinct pathways that lead to one event.

Imagine you have a bag of multicolored marbles. Without seeing inside, you want to predict the probability of pulling out a red marble. If the bag is divided into several smaller pouches, each containing a different mix of marble colors, the TPT allows you to consider the probability of choosing each pouch combined with the probability of then selecting a red marble from the chosen pouch.

In the context of the camera company example, we are interested in the event 'purchaser bought an extended warranty' (W). Since there are two types of cameras sold (basic B, and deluxe B'), the TPT helps us combine the separate scenarios: the probability of buying an extended warranty with a basic camera, and similarly with a deluxe camera. Mathematically, we express this as:

\[ P(W) = P(W | B)P(B) + P(W | B')P(B') \]

This formula encapsulates the essence of the Total Probability Theorem. It allows us to calculate the total probability of an event by summing up the probabilities of that event occurring under several distinct conditions.

Conditional Probability

Diving into Conditional Probability, we navigate the 'what if' of statistics and probability. This concept helps us understand the likelihood of an event occurring under the condition that another event has already happened. It offers a way to refine our predictions based on new information.

For instance, if someone prefers reading on sunny days and today's weather forecast shows a high chance of sunshine, we could more heavily predict them being found with a book in hand. This prediction is fundamentally different from our default expectation of their reading habits without the weather insight.

In our example regarding the video camera purchase, the conditional probability question is posed as the likelihood of having purchased a basic model given that the customer has already bought an extended warranty. Expressed in mathematical terms, we're looking for \( P(B|W) \). Here the vertical bar '|' stands for 'given that', so \( P(B|W) \) translates to the probability of event B occurring given that W has occurred. The calculation considers both the inherent probability of buying a basic model and the trends observed among those who buy warranties.

Probability Models

When we construct Probability Models, we are essentially building frameworks that allow us to predict or explain different outcomes within a specific context. These models are constructed based on known probabilities and conditions and can be thought of as probability trees or networks where each branch represents a potential outcome and is weighted by its likelihood.

In practice, probability models give us a systematic way to quantify uncertainty. For example, forecasting the weather involves creating probability models that consider pressure, temperature, and other meteorological data to predict rain, sunshine, or storms. Likewise, insurance companies use probability models to set premiums based on the risks associated with various life events.

In our scenario with camera sales and warranty purchases, the setting up of a probability model was implied when combining different events—buying a basic or deluxe model, and then opting for an extended warranty—under one umbrella to compute the overall probability of a customer opting for a warranty. This model then supports the use of Bayes' Theorem to reverse-engineer our understanding and find out which camera was likely bought if we know a warranty was purchased.