1. ## Sets and Probability

Prove:

$

P\{ (A \cup B) \cap C\} = P\{ A \cap C\} + P\{ B \cap C\} - P\{ A \cap B \cap C\}

$

2. The expression $P\left\{ {A \cap B} \right\} \cap C$ has no meaning.
It like adding a matrix to a number: meaningless!
So what are you working out?
What is the problem?

3. Originally Posted by kid funky fried
prove:

$
P\{ (A \cap B)\} {\color{red}\cap C} = P\{ A \cap C\} + P\{ B \cap C\} - P\{ A \cap B \cap C\}
$

So I did this:

$
\begin{gathered}
P(A \cup B) = P(A) + P(B) - P(A \cap B)theorum \hfill \\
by substitution, \hfill \\
(P(A) + P(B) - P(A \cap B)) \cap C = \hfill \\
P(A \cap C) + P(B \cap C) - P(A \cap B \cap C) \hfill \\
\end{gathered}
$
If it is no included inside of the probability term, $P\left\{(A\cap B)\right\}\cap C$ has no meaning, as Plato mentioned.

However, if it was $P\left\{(A\cap B)\cap C \right\}$, then we can work with it...maybe.

From what I'm understanding, I think you are looking for $P\left\{(A\cup B)\cap C \right\}$

This is the same as $P\left\{(A\cap C)\cup (B\cap C) \right\}={\color{red}P(A\cap C)+P(B\cap C)-P(A\cap B\cap C)}$

I hope this helps

--Chris

4. As I have pointed out to you, $P\left( {A \cap B} \right) \cap C$ is totally meaning less.
First $P\left( {A \cap B} \right)$ is a number while $C$ is a set in a probability space.
We do not intersect numbers with sets: it is meaningless.
I think that I have done this for you, did I not?
You cannot pove what you have written!

5. ## revsion

Hopefully, my revision has meaning.

I just want to make sure this is legit.

$
P(A \cap B) \cap P(C) = P(A \cap B \cap C)
$

Kid

6. Hi,
Originally Posted by kid funky fried
Hopefully, my revision has meaning.

I just want to make sure this is legit.

$
P(A \cap B) \cap P(C) = P(A \cap B \cap C)
$

Kid
Sorry to say, but this expression is not good either...
Again, $\cap$ is usually used for sets, whereas P(robabilities) are values, numbers...
You can't weigh apples and pears together when you buy fruits ^^

7. ## Thx

Thanks Moo, I think I understand now.
I will revise-again.

Kid

8. ## Have I seen the light?

$

\begin{gathered}
P\{ (A \cup B) \cap C\} = P(A \cap C) + P(B \cap C) - P(A \cap B \cap C) \hfill \\
Dist. \hfill \\
P\{ (A \cup B) \cap C\} = P\{ (A \cap C) \cup (B \cap C)\} \hfill \\
Theor. \hfill \\
P(A \cup B) = P(A) + P(B) - P(A \cap B) \hfill \\
I\_got, \hfill \\
P(A \cap C) + P(B \cap C) - P\{ (A \cap C) \cap (B \cap C)\} = \hfill \\
P(A \cap C) + P(B \cap C) - P\{ (A \cap B \cap C). \hfill \\
\end{gathered}

$

Now I understand Plato's disdain!
(I think)

9. Originally Posted by Moo
Hi,

Sorry to say, but this expression is not good either...
Again, $\cap$ is usually used for sets, whereas P(robabilities) are values, numbers...
You can't weigh apples and pears together when you buy fruits ^^

Moo, Do you think I could weigh apples and pears together if they are the same price?
( I do this sometimes, also!)lol

Thx for ur help. I appreciate it. I (obviously)did not realize that the intersection was the issue at hand.