MHB Conditional probability prove or disprove

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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