CFA CFA Level 1 probability question

# probability question

• Author
Posts
• 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

Viewing 2 reply threads
• You must be logged in to reply to this topic.