MHB Conditional probability prove or disprove

AI Thread Summary
The discussion centers on proving or disproving two statements about conditional probability. The first statement, P(A|B) = 1 - P(¬A|B), is shown to be true through logical derivation and the properties of probability measures. The second statement, P(A|B) = 1 - P(A|¬B), is disproven using a proof by contradiction, revealing a contradiction when assuming the statement is true. Participants also discuss the implications of P(B) being zero, noting that conditional probabilities become undefined in such cases. The conversation emphasizes the importance of understanding the definitions and properties of conditional probabilities in these proofs.
mathmari
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Hey! :o

Let $P$ be a probability measure on a $\sigma$-Algebra $\mathcal{A}$. I want to prove or disprove the following statements:
  1. $P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$
  2. $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$
I have done the following:
  1. We have that $B=\left (A\cap B\right )\cup \left (\overline{A}\cap B\right )$. Since the sets $\left (A\cap B\right )$ and $\left (\overline{A}\cap B\right )$ are disjoint we have that $P\left [\left (A\cap B\right )\cup \left (\overline{A}\cap B\right )\right ]=P\left (A\cap B\right )+P \left (\overline{A}\cap B\right )$.
    Therefore, we get $P(B)=P\left (A\cap B\right )+P \left (\overline{A}\cap B\right )$.

    We have that $P(A\mid B)=\frac{P(A\cap B)}{P(B)}\Rightarrow P(A\cap B)=P(A\mid B)P(B)$ and $P(\overline{A}\mid B)=\frac{P(\overline{A}\cap B)}{P(B)}\Rightarrow P(\overline{A}\cap B)=P(\overline{A}\mid B)P(B)$.

    So, we get $P(B)=P(A\mid B)P(B)+P(\overline{A}\mid B)P(B) \Rightarrow P(B)=P(B)\left [P(A\mid B)+P(\overline{A}\mid B)\right ] \overset{P(B)>0}{\Rightarrow }1=P(A\mid B)+P(\overline{A}\mid B) \Rightarrow P(A\mid B)=1-P(\overline{A}\mid B)$
    So, the statement is true.

    Is this correct? $$$$
  2. $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$

    From the definition of conditional probability then we get the statement $\frac{P(A\cap B)}{P(B)}=1-\frac{P(A\cap \overline{B})}{P(\overline{B})}$.

    For $A=\Omega$ we get $\frac{P(\Omega\cap B)}{P(B)}=1-\frac{P(\Omega\cap \overline{B})}{P(\overline{B})}\Rightarrow \frac{P( B)}{P(B)}=1-\frac{P( \overline{B})}{P(\overline{B})}\Rightarrow 1=1-1 \Rightarrow 1=0$, a contradiction.

    Therefore, this statement is in general not true.

    Is this correct?
 
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mathmari said:
1. $P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$

Hey mathmari!

What if $P(B)=0$? (Wondering)

mathmari said:
2. $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$

From the definition of conditional probability then we get the statement $\frac{P(A\cap B)}{P(B)}=1-\frac{P(A\cap \overline{B})}{P(\overline{B})}$.

How do we get that? (Wondering)
 
I like Serena said:
What if $P(B)=0$? (Wondering)

If $P(B)=0$ then the conditional probaibility $P(A\mid B)$ is not defined, is it? (Wondering)
I like Serena said:
How do we get that? (Wondering)

By definition we have that $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ and $P(A\mid \overline{B})=\frac{P(A\cap \overline{B})}{P(\overline{B})} $, or not? Therefore we get that $P(A\mid B)=1-P(A\mid \overline{B})$ is equivalent to $1-\frac{P(A\cap \overline{B})}{P(\overline{B})} $. Is this wrong? (Wondering)
 
mathmari said:
If $P(B)=0$ then the conditional probaibility $P(A\mid B)$ is not defined, is it? (Wondering)

Ah yes.
That's because $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, which is indeed undefined if $P(B)=0$.

mathmari said:
By definition we have that $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ and $P(A\mid \overline{B})=\frac{P(A\cap \overline{B})}{P(\overline{B})} $, or not?

Yes.

mathmari said:
Therefore we get that $P(A\mid B)=1-P(A\mid \overline{B})$ is equivalent to $1-\frac{P(A\cap \overline{B})}{P(\overline{B})} $. Is this wrong?

Well, I'm only aware of the complement rule $P(\overline U)=1-P(U)$, but it's not clear to me how this is applied. :confused:
 
I like Serena said:
Well, I'm only aware of the complement rule $P(\overline U)=1-P(U)$, but it's not clear to me how this is applied. :confused:

From the definition of the conditional probability we have that $P(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$ and $P(A\mid \overline{B})=\frac{\mathbb{P}(A\cap \overline{B})}{\mathbb{P}(\overline{B})}$.

Replacing these at $\mathbb{P}(A\mid B)=1-\mathbb{P}(A\mid \overline{B})$ we get $\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}=1-\frac{\mathbb{P}(A\cap \overline{B})}{\mathbb{P}(\overline{B})}$.

Isn't it correct? (Wondering)
 
mathmari said:
From the definition of the conditional probability we have that $P(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$ and $P(A\mid \overline{B})=\frac{\mathbb{P}(A\cap \overline{B})}{\mathbb{P}(\overline{B})}$.

Replacing these at $\mathbb{P}(A\mid B)=1-\mathbb{P}(A\mid \overline{B})$ we get $\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}=1-\frac{\mathbb{P}(A\cap \overline{B})}{\mathbb{P}(\overline{B})}$.

Isn't it correct? (Wondering)

Oh wait! You're doing a proof by contradiction!
I totally missed that.
Perhaps good to mention that. That is, 'suppose it is true, then...'. (Nerd)
 
I like Serena said:
Oh wait! You're doing a proof by contradiction!
I totally missed that.
Perhaps good to mention that. That is, 'suppose it is true, then...'. (Nerd)

Ah ok! Is the counterexample that I used correct? (Wondering)
 
mathmari said:
Ah ok! Is the counterexample that I used correct?

Yes. (Nod)
 
I like Serena said:
Yes. (Nod)

Great! Thank you! (Yes)
 

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