A small question

[Michael_CN]

Hi

I start self-learning statistics and probability and came across a small question. Could somebody help?

Let A, B, C be three events. Let B denotes the complement of B.

How can I represent P(B| A ^ C) in terms of P(A| B ^ C)

 

[Canuck]

i'm assuming ^=intersect
\(P(B|A^C)=1-P(B|A^C)=1-P(A^B^C)/P(A^C)\)
\(=1-[P(A^B^C)/P(A^C)]*(P(B^C)/P(B^C))=1-P(A| B ^ C)*P(B^C)/P(A^C)\)
is that what you want?

 

[Michael_CN]

Oops. Yes, ^ = interest. But the result is not exactly what I want.
The whole question is : Let A, B, C be three events such that P(A ^ C) and P(B ^ C) are both strictly between 0 and 1 and let B denotes the complement of B. Prove that: P(B|A^C) = P(A|B^C) * P(B|C) / P(A|C). I managed to do this proof.

The next question is to express P(A|C) in terms of P(A|B^C), P(A|B^C) and P(B|C)

Here I used the result from the prove question. So
P(A|C) = P(A|B^C) * P(B|C) / P(B|A^C). Comparing to the conditions given, the terms P(B|A^C) and P(A|B^C) are different. That is why I asked the simplified question. Your solution introduced the new terms P(A^C) and P(B^C)?

Anyway, thanks for your reply..!!! Further explanation will be appreciated.

 

[JulianT]

Here you go
\( P(A|C) = \frac{ P(A\cap C) }{ P(C) }=\frac{ P(A\cap C) }{ P(C) }\frac{ P(B\cap C) }{ P(B\cap C) } \)
\(=[ P(A \cap B \cap C) + P( A \cap \bar{B} \cap C )]\frac{ P(B|C) }{ P(B \cap C)}\)
\(= P(A| B \cap C)P(B|C)+ P(A | \bar{B} \cap C )P(B|C)\)

 

[Michael_CN]

Here you go
\( P(A|C) = \frac{ P(A\cap C) }{ P(C) }=\frac{ P(A\cap C) }{ P(C) }\frac{ P(B\cap C) }{ P(B\cap C) } \)
\(=[ P(A \cap B \cap C) + P( A \cap \bar{B} \cap C )]\frac{ P(B|C) }{ P(B \cap C)}\)
\(= P(A| B \cap C)P(B|C)+ P(A | \bar{B} \cap C )P(B|C)\)

Emm....I think something is wrong with the result. Where does \(P(A | \bar{B} \cap C )P(B|C)\) come from? Did you mistreat \(\bar{B}\) as \({B}\) so that you can merge \(P( A \cap \bar{B} \cap C )\) and \(P(B \cap C)\)?

 

[JulianT]

Indeed, my bad

 

[Canuck]

Hi

I start self-learning statistics and probability and came across a small question. Could somebody help?

Let A, B, C be three events. Let B denotes the complement of B.

How can I represent P(B| A ^ C) in terms of P(A| B ^ C)

give me the book name, page number and question. this will make it significantly easier to answer the question.

 

[Michael_CN]

Oh it's not from the book. It's just an exercise.

The original question is here.

 

[Canuck]

Oh it's not from the book. It's just an exercise.

The original question is here.

View attachment 5838

Is there not a section of reading material related to the question? It's usually the case that this section would have hints as to how to solve the question.

 

[Michael_CN]

I'm afraid not. It is a test exercise. I have no ideas where it comes from....