91Ó°ÊÓ

A large consumer goods company ran a television advertisement for one of its soap products. On the basis of a survey that was conducted, probabilities were assigned to the following events. \(B=\) individual purchased the product \(S=\) individual recalls seeing the advertisement \(B \cap S=\) individual purchased the product and recalls seeing the advertisement The probabilities assigned were \(P(B)=.20, P(S)=.40,\) and \(P(B \cap S)=.12\). a. What is the probability of an individual's purchasing the product given that the individual recalls seeing the advertisement? Does seeing the advertisement increase the probability that the individual will purchase the product? As a decision maker, would you recommend continuing the advertisement (assuming that the cost is reasonable)? b. Assume that individuals who do not purchase the company's soap product buy from its competitors. What would be your estimate of the company's market share? Would you expect that continuing the advertisement will increase the company's market share? Why or why not? c. The company also tested another advertisement and assigned it values of \(P(S)=.30\) and \(P(B \cap S)=.10 .\) What is \(P(B | S)\) for this other advertisement? Which advertisement seems to have had the bigger effect on customer purchases?

Step by step solution

01

Understanding Given Data

We need to identify the probabilities that are given in the problem and how they relate to each other. We have \(P(B) = 0.20\), which is the probability of purchasing the product, \(P(S) = 0.40\), which is the probability of recalling the advertisement, and \(P(B \cap S) = 0.12\), which is the probability of both purchasing and recalling the advertisement.

02

Calculate Conditional Probability

To find the probability of purchasing the product given that the individual remembers the advertisement, we use the conditional probability formula:\[ P(B|S) = \frac{P(B \cap S)}{P(S)} \]Substitute the values we have:\[ P(B|S) = \frac{0.12}{0.40} = 0.30 \]

03

Interpret the Conditional Probability

The probability of an individual purchasing the product given that they recall the advertisement is \(0.30\). Compare this with \(P(B) = 0.20\), to see if the advertisement influences the purchasing decision. Since \(0.30 > 0.20\), the advertisement increases the probability of purchase.

04

Recommendation on Advertising

Since seeing the ad increases the likelihood of purchase from \(0.20\) to \(0.30\), and assuming reasonable costs, continuing the advertisement seems beneficial. It effectively increases buyer conversion based on recall.

05

Estimating Market Share

Assuming the company's market consists of only those purchasing either their product or a competitor's, the company's estimated market share is \(P(B) = 0.20\). Hence, without considering absolute numbers, the market share is currently 20%.

06

Future Expectations with Continued Advertising

Since the advertisement increases the probability of purchase upon recall, continuing it could potentially increase market share by converting more people who remember the ad into buyers.

07

Calculate Conditional Probability for Second Ad

For the second advertisement, use the same formula:\[ P(B|S) = \frac{P(B \cap S)}{P(S)} \]Here, \(P(S) = 0.30\) and \(P(B \cap S) = 0.10\). Thus, \[ P(B|S) = \frac{0.10}{0.30} = 0.3333 \]

08

Comparison of Advertisements

The second ad results in a conditional probability of \(0.3333\), compared to \(0.30\) for the first ad. Despite having less recall, it slightly more effectively converts viewers into buyers. Thus, the second ad has a bigger positive effect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Market Share Analysis

Market share analysis is crucial for understanding a company's position in the marketplace relative to its competitors. Market share is a metric that represents the percentage of an industry's sales that a particular company controls. When estimating market share, we often assume that a limited set of options are available to consumers, such as purchasing a product from either the company or its competitors.
For instance, if the company's product has a probability of being bought at 20%, this implies a market share of 20% if we assume every potential buyer chooses between only this product and others. Understanding market share helps in crafting strategic decisions and assessing advertising's impact on exapanding this share.
The idea is to continually increase market share while reviewing where the brand stands in relation to competitors. This approach offers insights into how effective strategies, such as advertising, are in converting the potential audience into buyers. Evaluating this area is key to ensuring the company's product stays relevant and grows in its competitive landscape.

Advertising Effectiveness

Advertising effectiveness is the degree to which an advertisement achieves its intended performance goals. It is typically measured by observing an increase in purchase probability among those exposed to the ad. Conditional probability is a useful method for assessing this effectiveness, as it estimates the likelihood of purchasing the product given that the consumer has seen the advertisement.
We calculated this by using the formula for conditional probability: \[ P(B|S) = \frac{P(B \cap S)}{P(S)} \]By applying the given data, the probability of purchase rises from 20% to 30% upon recollection of the ad, which indicates that the ad has a positive impact on sales. This evidence suggests allocating resources towards maintaining or even increasing ad exposure might be worthwhile.
Furthermore, comparing different ads using similar metrics helps to determine which is more effective in making consumers more likely to buy the product. An ad generating a higher conditional probability indicates more effective customer conversion upon recall, thus better contributing to increased market share.

Decision Making in Business

Decision making in business involves evaluating the effectiveness of various strategies, such as advertising initiatives, to guide where to allocate resources for maximum impact. Businesses rely on statistical data, like market share and conditional probabilities, to make informed decisions. For example, when an ad increases the purchase probability from 20% to 30%, it reveals increased effectiveness, guiding decision-makers to continue investing in that ad.
In our case, comparing two different advertisements shows which has a stronger impact on consumer behavior. Deciding whether to continue or stop an ad involves assessing not only the purchase probability boost but also the costs versus benefits ratio. If one ad results in a conditional purchase probability of 0.333, marginally higher than another at 0.30, it might be wise to focus efforts there. These decisions are core to maintaining and enhancing a company's competitive edge.
Effective decision making is fundamental in optimizing profitability, growth, and brand presence in the market by leveraging advertising as a strategic tool to tip consumer choices towards the company's offerings.