 This topic has 2 replies, 3 voices, and was last updated Feb21 by asadrazanayani.

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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%.

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%

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


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