Study of analysts' forecasts. The Journal of Accounting Research (March 2008) published a study on the relationship between incentives and the degree of optimism among analysts' forecasts. Participants were analysts at either a large or small brokerage firm who made their forecasts early or late in the quarter. Also, some analysts were only concerned with making an accurate forecast, while others were also interested in their relationship with management. Suppose one of these analysts is randomly selected.

Consider the following events:

A = {The analyst is concerned only with making an accurate forecast.}

B = {The analyst makes the forecast early in the quarter.}

C = {The analyst is from a small brokerage firm.}

Describe the following events in terms of unions, intersections, and complements (e.g.,$\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{,}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{,}{\mathbf{A}}^{\mathbf{c}}$, etc.).

a. The analyst makes an early forecast and is concerned only with accuracy.

b. The analyst is not concerned only with accuracy.

c. The analyst is from a small brokerage firm or makes an early forecast.

d. The analyst makes a late forecast and is not concerned only with accuracy.

Probability is the chance or possibility that an event will occur, defined as the ratio of the number of positive instances to the total number of potential cases, assuming that all cases are equal. Consider the following event:

$\begin{array}{l}\mathbf{A}\mathbf{=}\mathbf{[}\mathbf{the}\mathbf{}\mathbf{analyst}\mathbf{}\mathbf{is}\mathbf{}\mathbf{only}\mathbf{}\mathbf{concerned}\mathbf{}\mathbf{with}\mathbf{}\mathbf{making}\mathbf{}\mathbf{an}\mathbf{}\mathbf{accurate}\mathbf{}\mathbf{forecast}\mathbf{]}\\ \mathbf{B}\mathbf{=}\mathbf{[}\mathbf{early}\mathbf{}\mathbf{in}\mathbf{}\mathbf{the}\mathbf{}\mathbf{quarter}\mathbf{,}\mathbf{the}\mathbf{}\mathbf{analyst}\mathbf{}\mathbf{makes}\mathbf{}\mathbf{the}\mathbf{}\mathbf{forecast}\mathbf{]}\\ \mathbf{C}\mathbf{=}\mathbf{[}\mathbf{the}\mathbf{}\mathbf{analystis}\mathbf{}\mathbf{form}\mathbf{}\mathbf{a}\mathbf{}\mathbf{large}\mathbf{}\mathbf{brokerage}\mathbf{}\mathbf{firm}\mathbf{]}\end{array}$

Two occurrences are complementary if one can happen only if the other does not happen. Consider the following complement event:

$\begin{array}{l}{\mathbf{A}}^{\mathbf{c}}\mathbf{=}\mathbf{\left[}\begin{array}{l}\mathbf{the}\mathbf{}\mathbf{analyst}\mathbf{}\mathbf{is}\mathbf{}\mathbf{not}\mathbf{}\mathbf{only}\mathbf{}\mathbf{concerned}\mathbf{}\mathbf{with}\mathbf{}\mathbf{accurate}\mathbf{,}\mathbf{also}\mathbf{}\mathbf{they}\mathbf{}\mathbf{are}\mathbf{}\mathbf{equally}\mathbf{}\mathbf{interested}\\ \mathbf{in}\mathbf{}\mathbf{the}\mathbf{}\mathbf{irrelationship}\mathbf{}\mathbf{with}\mathbf{}\mathbf{management}\mathbf{.}\end{array}\mathbf{\right]}\\ {\mathbf{B}}^{\mathbf{c}}\mathbf{=}\mathbf{[}\mathbf{the}\mathbf{}\mathbf{analyst}\mathbf{}\mathbf{makes}\mathbf{}\mathbf{a}\mathbf{}\mathbf{late}\mathbf{}\mathbf{forecast}\mathbf{.}\mathbf{]}\\ {\mathbf{C}}^{}\mathbf{=}\mathbf{[}\mathbf{the}\mathbf{}\mathbf{analystis}\mathbf{}\mathbf{form}\mathbf{}\mathbf{a}\mathbf{}\mathbf{large}\mathbf{}\mathbf{brokerage}\mathbf{}\mathbf{firm}\mathbf{.}\mathbf{]}\end{array}$

Consider the following event:

The analyst makes an early estimate and is just concerned with accuracy. This event represents a convergence of events. The analyst's primary interest is producing an accurate prognosis.

Therefore, the event is best described as follows:

$\mathbf{A}\mathbf{\cap }\mathbf{B}$

In this event, the analyst is concerned with more than just accuracy. However, the event's complement is that the analyst is just concerned with making an accurate forecast.

Therefore, the event is best described as follows:

${\mathbf{A}}^{\mathbf{c}}$

In this event, the analyst is from a small brokerage firm or makes an early forecast is the union of the event. The forecast is made by the analyst early in the quarter.

Therefore, the event is best described as follows:

$\mathbf{B}\mathbf{\cup }\mathbf{C}$

In this event, the analyst makes a late forecast and is not just concerned with accuracy but also with the intersection of the events. The analyst is concerned with more than just accuracy. These two occurrences are referred to as ${\mathbf{B}}^{\mathbf{c}}$and ${\mathbf{A}}^{\mathbf{c}}$.

Therefore, the event is best described as follows:

${\mathbf{B}}^{\mathbf{c}}\mathbf{\cap }{\mathbf{A}}^{\mathbf{c}}$