Question

An electronics store sells two different brands of DVD players. The store reports that \(30 \%\) of customers purchasing a DVD choose Brand \(1 .\) Of those that choose Brand \(1,20 \%\) purchase an extended warranty. Consider the chance experiment of randomly selecting a customer who purchased a DVD player at this store. a. One of the percentages given in the problem specifies an unconditional probability, and the other percentage specifies a conditional probability. Which one is the conditional probability, and how can you tell? b. Suppose that two events \(B\) and \(E\) are defined as follows: \(B=\) selected customer purchased Brand 1 \(E=\) selected customer purchased an extended warranty Use probability notation to translate the given information into two probability statements of the form \(P(\underline{ })=\) probability value.

Short answer

Part a: The unconditional probability is the \(30% \) chance of customers choosing Brand 1. The conditional probability is the \(20% \) chance of a customer purchasing an extended warranty given that they have already chosen Brand 1. Part b: The probability of a customer buying Brand 1 (\(P(B) = 0.30 \)) and the probability of a customer purchasing an extended warranty given they bought Brand 1 (\(P(E | B) = 0.20\)).

Step-by-step solution

Understand the Concept of Conditional and Unconditional Probabilities

An unconditional probability is the chance of an event happening irrespective of any other events. In this exercise, the probability of a client choosing Brand 1 is unconditional. In contrast, a conditional probability is the probability of event A happening given that another event B has already occurred. In this case, the conditional probability is the probability that a customer purchases an extended warranty given that they have chosen Brand 1.

Identify the Unconditional and Conditional Probabilities

From the problem, the unconditional probability is the \(30%\) chance of customers choosing Brand 1. The conditional probability is the \(20%\) chance of a customer purchasing an extended warranty given that they have already chosen Brand 1.

Translate the Information into Probability Notation

Translate the information into probability notation as stated in the problem. Let event \(B\) represent a customer purchasing Brand 1, and let \(E\) represent a customer purchasing an extended warranty. Therefore, the probability of a customer buying Brand 1 (\(B\)) is \(P(B) = 0.30\) (or \(30% \)), and the probability of a customer purchasing an extended warranty given they bought Brand 1 (\(E | B\)) is \(P(E | B) = 0.20\) (or \(20% \)).

Key concepts

Unconditional Probability

Unconditional probability, also known as marginal or simple probability, is the basic concept of how likely an event is to occur without any conditions or restrictions applied. It's like flipping a coin and saying the probability of getting heads is 50%, regardless of any previous coin flips. In the exercise described above, the 30% figure—that is, the chance of a customer choosing Brand 1—is an example of an unconditional probability. This percentage doesn't depend on any other event; it simply reflects the overall likelihood of customers picking this brand from all the DVD players available in the store.

Understanding unconditional probability is vital in making standalone predictions and serves as a building block for more complex probability concepts, such as conditional probability and joint probability. When dealing with more complex scenarios where events might influence one another, unconditional probability provides a baseline from which conditional probabilities can be compared and analyzed.

Probability Notation

Probability notation is the language used to succinctly describe the likelihood of events. In mathematical terms, probabilities are generally expressed as a number between 0 and 1, with 0 meaning an event will definitely not occur, and 1 signifying an event is certain. Fractions, decimals, and percentages are other ways to express probability values. We use the capital letter 'P' to denote probability followed by the event of interest in parentheses. For instance, if we have an event A, the probability of A occurring is written as \( P(A) \).

In the context of our DVD player example, probability notation helps us translate textual descriptions into a mathematical framework, which is crucial for further calculations or inferences. With this notation, we can express the chance of a customer purchasing Brand 1 as \( P(B) = 0.30 \), and similarly, this notation also allows us to define the chance of a customer not purchasing Brand 1 as \( P(eg B) \), where \( eg B \) indicates the event in which Brand 1 is not chosen.

Extended Warranty Probability

Extended warranty probability is a specific type of conditional probability. It tackles the likelihood of a customer purchasing an extended warranty but only among those who have already made a specific purchase—in this case, those who bought Brand 1 DVD players. Here, we're not considering all customers, but just a subsection of them. To find this, we look at what proportion of the customers who chose Brand 1 went on to buy an extended warranty. This probability is expressed as \( P(E | B) \), where 'E' is the event of purchasing an extended warranty, and 'B' is the event of purchasing Brand 1. The vertical bar '|' is read as 'given'.

The probability of someone buying an extended warranty given they bought a Brand 1 DVD player—\( P(E | B) = 0.20 \) or 20% in this case—is helpful for the store in understanding how their additional offerings like extended warranties are performing among the different customer segments. Knowing this can drive decisions on marketing strategies or product pair offerings to enhance sales and customer satisfaction.