probability question

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

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%.