B Why did it suddenly become subtractive? (Example of Bayes’ Theorem)

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The discussion revolves around the application of Bayes' Theorem in a financial context, specifically analyzing the relationship between CEO replacements and stock price increases. Participants clarify the calculation of probabilities, particularly the use of 1 in the formula, which represents certainty in probability terms. The importance of parentheses in mathematical expressions is emphasized to avoid confusion in calculations. The conversation highlights the need to understand both the probabilities of stock price increases and the conditions under which CEOs are replaced. Ultimately, the goal is to accurately determine the probability that a company's stock will increase by more than 5% after a CEO replacement.
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From https://corporatefinanceinstitute.com/resources/data-science/bayes-theorem/#:~:text=Formula for Bayes' Theorem&text=P(A|B) –,given event A has occurred

Example of Bayes’ Theorem
Imagine you are a financial analyst at an investment bank. According to your research of publicly-traded companies, 60% of the companies that increased their share price by more than 5% in the last three years replaced their CEOs during the period.

At the same time, only 35% of the companies that did not increase their share price by more than 5% in the same period replaced their CEOs. Knowing that the probability that the stock prices grow by more than 5% is 4%, find the probability that the shares of a company that fires its CEO will increase by more than 5%.

Before finding the probabilities, you must first define the notation of the probabilities.

P(A) – the probability that the stock price increases by 5%
P(B) – the probability that the CEO is replaced
P(A|B) – the probability of the stock price increases by 5% given that the CEO has been replaced
P(B|A) – the probability of the CEO replacement given the stock price has increased by 5%.
Using the Bayes’ theorem, we can find the required probability:

Sample Calculation

P(A l B) = 0.60 x 0.04/0.60 x 0.04 + 0.35 x (1 - 0.04) = 0.067 or 6.67%

Thus, the probability that the shares of a company that replaces its CEO will grow by more than 5% is 6.67%.

Sorry but notice the bold numbers, how did (1 - 0.04) appear there? I can't find 1 mentioned in the question? English is not my native language.
 
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Hi,

can you fix the errors?

0.60 x 0.04/0.60 x 0.04 + 0.35 + (1 - 0.04) =1.3116

so something isn't right.

1-0.04 is the probability that the stock prices do NOT grow by more than 5%

##\ ##
 
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BvU said:
Hi,

can you fix the errors?

0.60 x 0.04/0.60 x 0.04 + 0.35 + (1 - 0.04) =1.3116

so something isn't right.

1-0.04 is the probability that the stock prices do NOT grow by more than 5%

##\ ##

Yeah sorry, it supposed to be 0.35 x (multiply) (1-0.04).

So, 1 - 0.04 is 0.96 which is more likely to happen than 0.04? Is that correct?

Why did we used 1 instead of any numbers? Is that part of the formula? It mentioned something like binary variable, something like that.
 
One step at atime ?

0.60 x 0.04/0.60 x 0.04 + 0.35 * (1 - 0.04) = 0.3376

But we are getting there ...

:wink: sorrry to be so obnoxious ...


##\ ##
 
BvU said:
One step at atime ?

0.60 x 0.04/0.60 x 0.04 + 0.35 * (1 - 0.04) = 0.3376

But we are getting there ...

:wink: sorrry to be so obnoxious ...


##\ ##

Ok, one step at a time. I am still confused why we should use 1 but not other number. Especially the 0.35*(1 - 0.04).
 
What I mean is: Do not forget the brackets !

0.60 x 0.04/(0.60 x 0.04 + 0.35 * (1 - 0.04)) = ...

Bayes:
$$P(A|B) = {P(B|A)\,P(A)\over P(B)}$$compare that to the much more legible
$$P(A|B)={0.60\times 0.04\over 0.60\times 0.04+0.35\times (1-0.04)}$$
knowing
  • 35% of the companies that did not increase their share price by more than 5% in the same period replaced their CEOs
  • 60% of the companies that increased their share price by more than 5% in the last three years replaced their CEOs
In short: what is ##{ 0.60\times 0.04+0.35\times (1-0.04)}## ?


BvU said:
1-0.04 is the probability that the stock prices do NOT grow by more than 5%
We use the number 1 for a probability that is a certainty (100%).
For example $$P(A)+P(\neg A)=1$$ (##\neg A ## means NOT ##A## )

So in the denominator we expect to see ##P(B)## appearing: the probability the ceo is replaced

##\ ##
 
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wonderingchicken said:
Ok, one step at a time. I am still confused why we should use 1 but not other number. Especially the 0.35*(1 - 0.04).
But it is much easier to begin answering your question if it is correctly stated. A computer (and many people) will take your equation literally. It doesn't cost you anything to put parenthesis around the entire denominator.
0.60 * 0.04/(0.60 * 0.04 + 0.35 * (1 - 0.04))
 
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BvU said:
What I mean is: Do not forget the brackets !

0.60 x 0.04/(0.60 x 0.04 + 0.35 * (1 - 0.04)) = ...

Bayes:
$$P(A|B) = {P(B|A)\,P(A)\over P(B)}$$compare that to the much more legible
$$P(A|B)={0.60\times 0.04\over 0.60\times 0.04+0.35\times (1-0.04)}$$
knowing
  • 35% of the companies that did not increase their share price by more than 5% in the same period replaced their CEOs
  • 60% of the companies that increased their share price by more than 5% in the last three years replaced their CEOs
In short: what is ##{ 0.60\times 0.04+0.35\times (1-0.04)}## ?



We use the number 1 for a probability that is a certainty (100%).
For example $$P(A)+P(\neg A)=1$$ (##\neg A ## means NOT ##A## )

So in the denominator we expect to see ##P(B)## appearing: the probability the ceo is replaced

##\ ##

Oh, so the first P (B| A) P (A) is did increase their share by 5% and the second P (B| A) P (A) is did NOT increase their share by 5%, correct? If NOT, we have to subtract 1 with the first P(A)?
 
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Yes. Spelling it out:

We are after
  • ##P(A|B)##, the probability that the shares of a company that fires its CEO will increase by more than 5%
knowing
  • ##P(B|A) = 0.60##, the probability of the CEO replacement given the stock price has increased by 5%
  • ##P(A)=0.04##, the probability that the stock prices grow by more than 5%
and Bayes' formula tells us
$$P(A|B) = {P(B|A)\,P(A)\over P(B)}$$so all we still need is
  • ##P(B)##, the probability that the CEO is replaced
For this last one we use$$P(B) = P(B|A)\,P(A)+P(B|\neg A)\,P(\neg A)$$
knowing
  • ##P(B|\neg A) = 0.35##, 35% of the companies that did not increase their share price by more than 5% replaced their CEOs
  • ##P(\neg A) = 1-0.04##, the probability that the stock prices do not grow by more than 5%
making use of the fact that stock prices either grow more than 5% or not grow more than 5%:

##\ ##
 
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BvU said:
Yes. Spelling it out:

We are after
  • ##P(A|B)##, the probability that the shares of a company that fires its CEO will increase by more than 5%
knowing
  • ##P(B|A) = 0.60##, the probability of the CEO replacement given the stock price has increased by 5%
  • ##P(A)=0.04##, the probability that the stock prices grow by more than 5%
and Bayes' formula tells us
$$P(A|B) = {P(B|A)\,P(A)\over P(B)}$$so all we still need is
  • ##P(B)##, the probability that the CEO is replaced
For this last one we use$$P(B) = P(B|A)\,P(A)+P(B|\neg A)\,P(\neg A)$$
knowing
  • ##P(B|\neg A) = 0.35##, 35% of the companies that did not increase their share price by more than 5% replaced their CEOs
  • ##P(\neg A) = 1-0.04##, the probability that the stock prices do not grow by more than 5%
making use of the fact that stock prices either grow more than 5% or not grow more than 5%:

##\ ##
Thank you so much, Sir.
 
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