Question

A retailer sells two styles of highpriced digital video recorders (DVR) that experience indicates are in equal demand. (Fifty percent of all potential customers prefer style \(1,\) and \(50 \%\) favor style \(2 .\) ) If the retailer stocks four of each, what is the probability that the first four customers seeking a DVR all purchase the same style?

Short answer

Answer: The probability is 1/8.

Step-by-step solution

Calculate the total number of ways the customers can choose DVRs

Since there are two styles of DVRs and equal demand for both styles, we can represent the choices of the customers using a binary sequence, where 1 represents Style 1 and 0 represents Style 2. Thus, if a customer chooses Style 1, the sequence will have a 1, and if a customer chooses Style 2, the sequence will have a 0. As there are four customers, the sequence will have four places. A binary sequence with 4 places can have a total of \(2^4=16\) different combinations.

Calculate the number of ways the four customers can all choose the same style

There are only two ways the first four customers can all buy the same style of DVR: 1. All four customers buy Style 1. 2. All four customers buy Style 2.

Calculate the probability

Now that we have the total number of ways the customers can choose, and the number of ways they all choose the same style, we can calculate the probability. Number of desired outcomes (all choosing the same style) = 2 Total number of outcomes = 16 Probability = \(\frac{\text{Number of desired outcomes}}{\text{Total number of outcomes}} = \frac{2}{16}\) After simplifying, the probability is Probability = \(\frac{1}{8}\) So the probability that the first four customers seeking a DVR all purchase the same style is \(\frac{1}{8}\).

Key concepts

Binary sequence

In probability and combinatorics, a binary sequence is a sequence consisting of two possible values, typically 0 and 1. These sequences are useful for representing different scenarios or outcomes with limited options. In our exercise, the binary sequence represents customer preferences for digital video recorders (DVRs). Depending on their choice, we assign either 0 for Style 2 or 1 for Style 1, given the two styles available.

With four customers choosing between two styles, the sequence has four positions, allowing for a total of \(2^4 = 16\) possible binary combinations. These combinations depict all the ways customers can select between the two styles. Understanding how to create and use binary sequences is essential for calculating various probabilities and outcomes in similar scenarios.

Digital video recorders

Digital video recorders (DVRs) are devices used to record, store, and playback digital video content. They have revolutionized how people watch television by enabling time-shifting capabilities, which allow viewers to record shows and watch them at their convenience. In the context of the problem, the retailer offers two high-priced styles of DVRs, indicating that both styles have distinct features and appeal.

The preference for each style is evenly divided among potential customers at 50% for Style 1 and 50% for Style 2. This scenario provides a controlled environment to analyze the purchasing behavior of customers and to calculate probabilities based on customers' selections. It's common for advanced electronics like DVRs to have different models or styles, catering to specific customer needs and preferences.

Equal demand

When customers exhibit equal demand for two products, it means that both products are equally likely to be chosen by any given customer. In our problem, this 50-50 distribution between the two styles of DVRs simplifies the probability calculation by ensuring that each customer's choice is independent and identically distributed.

This characteristic forms an essential part of our probability model. It simplifies assumptions and calculations since we don't have to account for any inherent bias or preference towards one of the styles. With equal demand, a customer's decision to buy a specific DVR is purely a matter of chance, mirroring a fair coin flip between two equally appealing choices.

Probability calculation

Probability calculation is about determining the likelihood of a specific event occurring. In our problem, we want to find the probability that the first four customers purchase the same style of DVR.

First, we compute the total number of possible buying sequences for the DVRs. This is represented by the 16 possible outcomes for the four customers, based on binary sequences.

• There's one scenario where all four customers choose Style 1: \(1111\).
• Another scenario occurs when all choose Style 2: \(0000\).

This means there are 2 desirable outcomes leading to all picking the same style out of 16 total possibilities.
Thus, the probability is calculated as \(\frac{2}{16} = \frac{1}{8}\). The result signifies that there's a 1 in 8 chance that all four customers will end up choosing the same DVR style. Probability calculations are key in understanding likely trends and patterns in decision-making scenarios like this one.