A market-research group specializes in providing assessments...

A market-research group specializes in providing assessments of the prospects of sites for new children's toy stores in shopping centers. The group assesses prospects as good, fair, or poor. The records of assessments made by this group were examined, and it was found that for all stores that had annual sales over $$\$ 1,000,000$$, the assessments were good for $70 \%$, fair for $20 \%$, and poor for $10 \%$. For all stores that turned out to be unsuccessful, the assessments were good for $20 \%$, fair for $30 \%$, and poor for $50 \%$. It is known that $60 \%$ of new clothing stores are successful and $40 \%$ are unsuccessful. a. For a randomly chosen store, what is the probability that prospects will be assessed as good? b. If prospects for a store are assessed as good, what is the probability that it will be successful? c. Are the events "prospects assessed as good" and "store is successful" statistically independent? d. Suppose that five stores are chosen at random. What is the probability that at least one of them will be successful?

A market-research group specializes in providing assessments of the prospects of sites for new children's toy stores in shopping centers. The group assesses prospects as good, fair, or poor. The records of assessments made by this group were examined, and it was found that for all stores that had annual sales over $$\$ 1,000,000$$, the assessments were good for $70 \%$, fair for $20 \%$, and poor for $10 \%$. For all stores that turned out to be unsuccessful, the assessments were good for $20 \%$, fair for $30 \%$, and poor for $50 \%$. It is known that $60 \%$ of new clothing stores are successful and $40 \%$ are unsuccessful.
a. For a randomly chosen store, what is the probability that prospects will be assessed as good?
b. If prospects for a store are assessed as good, what is the probability that it will be successful?
c. Are the events "prospects assessed as good" and "store is successful" statistically independent?
d. Suppose that five stores are chosen at random. What is the probability that at least one of them will be successful?

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B) and is fundamental in the analysis of problems where outcomes depend on previous conditions or evaluations, such as assessments influenced by store success or failure.

Law of Total Probability

The law of total probability is a rule that allows the computation of the overall probability of an event by considering all the mutually exclusive ways in which the event can occur. In problems involving different groups (for example, successful and unsuccessful stores), it is used to combine the probabilities weighted by the likelihood of each group.

Bayes’ Theorem

Bayes’ Theorem is a method to update the probability estimate for an event based on new information. It relates the conditional and marginal probabilities, and is particularly useful in determining the probability of a store being successful given a certain assessment, by reversing the conditioning from the known data.

Statistical Independence

Statistical independence between two events means that the occurrence of one does not affect the probability of the other. Understanding whether assessments and actual success are independent is crucial to determine if the events interact or if one provides no new information about the other.

Complement Rule

The complement rule is a basic concept in probability stating that the probability of an event occurring is one minus the probability of the event not occurring. This rule is often used in calculating the probability of at least one event happening in a series of trials, such as determining the likelihood of at least one store being successful among several.