Question

An investor in Apple is worried the latest management earnings forecast is too aggressive and the company will fall short. His favorite analyst that covers Apple is going to release his report on Apple the week before the earnings announcement. Report stands for the analyst's report, and Forecast stands for the earnings announcement. Refer to Exhibit 4-7. What is the probability the analyst issued a good report given Apple's earnings announcement was below the forecast?

Answer 1

Answer:

$P(GoodR/Below Forecast)= \frac{P(GoodR n Below Forecast)}{P(Below Forecast)}= \frac{0.02}{0.49} = 0.04$

Step-by-step explanation:

Hello!

Given the probability information about analyst's report (Good, Medium,  and Bad) and the earnings announcement (Forecast),  you have to calculate the probability that the analyst issued a good report, given that the earnings announcement was "bellow the forecast".

Symbolically: P(Good Report/Below Forecast)

This is a conditional probability, a little reminder:

If you have the events A and B, that are not independent, the probability of A given that B has already happened can be calculated as:

$P(A/B)= \frac{P(AnB)}{P(B)}$

Where P(A∩B) is the intersection between the two events and P(B) represents the marginal probability of ocurrence of B.

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Using that formula:

$P(Good Report/Below Forecast)= \frac{P(Good Report n Below Forecast)}{P(Below Forecast)}$

As you can see you have to calculate the value of the probability for the intersection between "Good report" and "Below Forecast" and the probability for P(Below Forecast)

Using the given probability values you can clear the value of the intersection:

$P(BelowForecast/Good Report)= \frac{P(Good Report n Below Forecast)}{P(Good Report)}$

P(Good Report ∩ Below Forecast)= P(BelowForecast/GoodReport)*P(Good Report)= 0.1*0.2= 0.02

Now the probability of an earnings announcement being "Below Forecast" is marginal, that is if you were to arrange all possible outcomes in a contingency table this probability will be in the marginal sides of the table:

                               Below Forecast

Good Report          P(GoodR∩BelowF)

Medium Report      P(MediumR∩BelowF)

Bad Report             P(BadR∩BelowF)

Total                        P(Below Forecast)

Then P(BelowF)=P(GoodR∩BelowF)+P(MediumR∩BelowF)+P(BadR∩BelowF)

You can clear the two missing probabilities from the remaining information:

$P(BelowForecast/MediumR)= \frac{P(BelowForecast n MediumR}{P(MediumR)}$

P(BelowF∩MediumR)= P(BelowF/MediumR)*P(MediumR)= 0.4*0.5= 0.2

$P(BelowForecast/BadR)= \frac{P(BelowForecastnBadR)}{P(BadR)}$

P(BelowF∩BadR)= P(BelowF/BadR)*P(BadR)= 0.9*0.3= 0.27

Now you can calculate the probability of the earning announcement being below forecast:

P(BelowF)=P(GoodR∩BelowF)+P(MediumR∩BelowF)+P(BadR∩BelowF)

P(BelowF)= 0.02+0.2+0.27= 0.49

And finally the asked probability is:

$P(GoodR/Below Forecast)= \frac{P(GoodR n Below Forecast)}{P(Below Forecast)}= \frac{0.02}{0.49} = 0.04$

I hope this helps!