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

[Rev. Cheeseman]

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.

 

[BvU]

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%

$\$

 

[Rev. Cheeseman]

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.

 

[BvU]

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 ...

sorrry to be so obnoxious ...


$\$

 

[Rev. Cheeseman]

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 ...

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).

 

[BvU]

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 ${ 0.60\times 0.04+0.35\times (1-0.04)}$ ?

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

$\$

 

[FactChecker]

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))

 

[Rev. Cheeseman]

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)?

 

[BvU]

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%:

$\$

 

[Rev. Cheeseman]

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.