The accompanying table shows, for 1,000 forecasts of...

The accompanying table shows, for 1,000 forecasts of earnings per share made by financial analysts, the numbers of forecasts and outcomes in particular categories (compared with the previous year). $$ \begin{array}{lcrc} \hline &{}{}{\text { Forecast }} \\ \text { Outcome } & \text { Improvement } & \text { About the } \\ \text { Same } & \text { Worse } \\ \hline \text { Improvement } & 210 & 82 & 66 \\ \text { About the same } & 106 & 153 & 75 \\ \text { Worse } & 75 & 84 & 149 \\ \hline \end{array} $$ a. Find the probability that if the forecast is for a worse performance in earnings, this outcome will result. b. If the forecast is for an improvement in earnings, find the probability that this outcome fails to result.

The accompanying table shows, for 1,000 forecasts of earnings per share made by financial analysts, the numbers of forecasts and outcomes in particular categories (compared with the previous year).
$$
\begin{array}{lcrc}
\hline &{}{}{\text { Forecast }} \\
\text { Outcome } & \text { Improvement } & \text { About the } \\
\text { Same } & \text { Worse } \\
\hline \text { Improvement } & 210 & 82 & 66 \\
\text { About the same } & 106 & 153 & 75 \\
\text { Worse } & 75 & 84 & 149 \\
\hline
\end{array}
$$
a. Find the probability that if the forecast is for a worse performance in earnings, this outcome will result.
b. If the forecast is for an improvement in earnings, find the probability that this outcome fails to result.

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Key Concepts

Marginal Totals and Relative Frequencies

Marginal totals are the sums of rows or columns in a contingency table, which offer insights into the overall distribution of each variable. Relative frequencies derived from these marginals are crucial for computing probabilities related to specific events or conditions in the data.

Complementary Probability

Complementary probability refers to the probability of the complement event, that is, the event not occurring. It is determined by subtracting the probability of the event from one and is essential when determining the likelihood of outcomes that do not meet the forecast or expected condition.

Contingency Table

A contingency table is a two-dimensional array used to display the frequency distribution of two categorical variables. It helps in organizing data to show the relationship (or dependency) between variables, such as forecasts and outcomes in this example, and serves as a basis for calculating various probabilities.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has occurred. It is calculated by taking the number of favorable outcomes under a specific condition divided by the total number of cases satisfying that condition, often using data provided in contingency tables.

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