 This topic has 3 replies, 4 voices, and was last updated Oct238:48 am by pcunniff.

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Up::11
Hello! I picked 26% and got it wrong. Does anyone know why its 69%? Thanks!
An analyst expects that 20% of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 90% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 10% chance that the ratio will be negative. The analyst randomly selects a company with a negative ratio. Based on Bayes’ theorem, the updated probability that the company will experience a decline is:
A)
26%.
B)
69%.
C)
18%.

Up::5
Hi @pcunniff – this is a popular test topic.
Here’s my workings:
P(decline) = 0.2
P (not decline) = 0.8
P(negative ratio  decline) = 0.9
P(negative ratio  not decline) = 0.1
So we want to look for P(decline  negative ratio), i.e. probability of earnings decline given negative ratio.
Therefore, P(decline  negative ratio) = [P( negative ratio  decline) x P(decline)] / P(negative ratio)
= 0.9 * 0.2 / [ P(negative ratio  decline) * P(decline) + P(negative ratio  not decline) * P(not decline)]
= (0.9*0.2) / [ 0.9*0.2 + 0.1*0.8]
= 69.23%

Up::5
I personally like to draw out a diagram, so
Ber..jpg
We have been asked to calculate the P(decline assuming ve ratio).
The key realization is: P(ve ratio and decline) = P(decline and ve ratio)
Now: P(ve ratio and decline) = P(ve ratio assuming decline) x P(decline) = P(decline and ve ratio) = P(decline assuming ve ratio) x P(ve ratio)
Therefore: P(ve ratio assuming decline) x P(decline) = P(decline assuming ve ratio) x P(ve ratio)
We can calculate P(ve ratio) as
P(ve ratio) = (P(ve ratio assuming decline) x P(decline)) + (P(ve ratio assuming not decline) x P(not decline))
We can now replace
P(decline assuming ve ratio) = (P(ve ratio assuming decline) x P(decline)) / (P(ve ratio assuming decline) x P(decline)) + (P(ve ratio assuming not decline) x P(not decline))
P(decline assuming ve ratio) = 0.9 x 0.2 / ((0.9 x 0.2) + (0.8 x 0.1))
P(decline assuming ve ratio) = 0.6923076923

Up::0
As knowledge from bloxd io, to determine the updated probability that the company will experience a decline, we can use Bayes’ theorem. Let’s denote the events as follows:
A: Company experiences a decline in earnings
B: The selected company has a negative ratioWe are given the following information:
P(A) = 20% = 0.20 (probability of a company experiencing a decline)
P(BA) = 90% = 0.90 (probability of a negative ratio given a decline)
P(Bnot A) = 10% = 0.10 (probability of a negative ratio given no decline)We want to find P(AB), the updated probability that the company will experience a decline given that the ratio is negative.
Using Bayes’ theorem:
P(AB) = (P(BA) * P(A)) / P(B)
To calculate P(B), we need to consider the probabilities of both scenarios (decline and no decline):
P(B) = P(BA) * P(A) + P(Bnot A) * P(not A)
P(not A) represents the probability of not experiencing a decline, which is the complement of P(A).
P(not A) = 1 – P(A) = 1 – 0.20 = 0.80
Now we can substitute the values into the equation:
P(B) = (0.90 * 0.20) + (0.10 * 0.80) = 0.18 + 0.08 = 0.26
Finally, we can calculate P(AB):
P(AB) = (P(BA) * P(A)) / P(B) = (0.90 * 0.20) / 0.26 â‰ˆ 0.692
So, the updated probability that the company will experience a decline, given that the ratio is negative, is approximately 69%. Therefore, the correct answer is B) 69%.


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