CFA CFA Level 1 probability question

probability question

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    • pcunniff
      Participant
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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%.

    • Sophie Macon
      Keymaster
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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%

    • asadrazanayani
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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

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