CFA CFA Level 1 probability question

probability question

  • This topic has 3 replies, 4 voices, and was last updated Oct-238:48 am by pcunniff.
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    • Avatar of pcunniffpcunniff
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        • CFA Level 1
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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%.

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        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
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        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

      • Avatar of samuelhiltonsamuelhilton
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          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 ratio

          We are given the following information:

          P(A) = 20% = 0.20 (probability of a company experiencing a decline)
          P(B|A) = 90% = 0.90 (probability of a negative ratio given a decline)
          P(B|not A) = 10% = 0.10 (probability of a negative ratio given no decline)

          We want to find P(A|B), the updated probability that the company will experience a decline given that the ratio is negative.

          Using Bayes’ theorem:

          P(A|B) = (P(B|A) * P(A)) / P(B)

          To calculate P(B), we need to consider the probabilities of both scenarios (decline and no decline):

          P(B) = P(B|A) * P(A) + P(B|not 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(A|B):

          P(A|B) = (P(B|A) * P(A)) / P(B) = (0.90 * 0.20) / 0.26 ≈ 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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