Before books aimed at preschool children are marketed,...

Before books aimed at preschool children are marketed, reactions are obtained from a panel of preschool children. These reactions are categorized as favorable, neutral, or unfavorable. Subsequently, book sales are categorized as high, moderate, or low, according to the norms of this market. Similar panels have evaluated 1,000 books in the past. The accompanying table shows their reactions and the resulting market performance of the books. $$ \begin{array}{lccc} \hline & & {\text { Panel Reaction }} \\ \hline \text { Sales } & \text { Favorable } & \text { Neutral } & \text { Unfavorable } \\ \hline \text { High } & 173 & 101 & 61 \\ \text { Moderate } & 88 & 211 & 70 \\ \text { Low } & 42 & 113 & 141 \\ \hline \end{array} $$ a. If the panel reaction is favorable, what is the probability that sales will be high? b. If the panel reaction is unfavorable, what is the probability that sales will be low? c. If the panel reaction is neutral or better, what is the probability that sales will be low? d. If sales are low, what is the probability that the panel reaction was neutral or better?

Before books aimed at preschool children are marketed, reactions are obtained from a panel of preschool children. These reactions are categorized as favorable, neutral, or unfavorable. Subsequently, book sales are categorized as high, moderate, or low, according to the norms of this market. Similar panels have evaluated 1,000 books in the past. The accompanying table shows their reactions and the resulting market performance of the books.
$$
\begin{array}{lccc}
\hline & & {\text { Panel Reaction }} \\
\hline \text { Sales } & \text { Favorable } & \text { Neutral } & \text { Unfavorable } \\
\hline \text { High } & 173 & 101 & 61 \\
\text { Moderate } & 88 & 211 & 70 \\
\text { Low } & 42 & 113 & 141 \\
\hline
\end{array}
$$
a. If the panel reaction is favorable, what is the probability that sales will be high?
b. If the panel reaction is unfavorable, what is the probability that sales will be low?
c. If the panel reaction is neutral or better, what is the probability that sales will be low?
d. If sales are low, what is the probability that the panel reaction was neutral or better?

Key Concepts

Conditional Probability

This concept describes the likelihood of an event occurring given that another event has already occurred. In the context of this problem, it is used to calculate the probability of certain sales outcomes given specific panel reactions by focusing on the subset of data where the condition holds and then determining the ratio of favorable outcomes within that subgroup.

Contingency Table

A contingency table is a type of table in a matrix format that displays the frequency distribution of variables. It is used to show the relationship between two categorical variables by summarizing data in rows and columns, such as panel reactions and sales categories, which facilitates the calculation of conditional and joint probabilities.

Marginal Probability

Marginal probability refers to the probability of an event occurring, irrespective of the outcome of another variable. In a contingency table, the marginal probability is found by summing the frequencies across rows or columns to give the total occurrence of a particular outcome, which is then used as the denominator in conditional probability calculations.

Joint Frequency Distribution

This concept involves the counts of occurrences where two or more events happen simultaneously. In the problem, the joint frequency distribution represents the number of books that received each combination of panel reaction and sales performance, which forms the basis for calculating both marginal and conditional probabilities.

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