Conditional Probability problem

Let S be a sample space, with A is a subset S and B is a subset S. If P(A) = .6 what can be said about, P(A ∩ B),

when

(a) A and B are mutully exclusive?

(b) A is a subset B

(c) B is a subset A

(d) A′ is a subset B′

(e) A is a subset B′

Not entirely sure how to approach this problem...let me know if the first is on the right track.

a) If A and B are mutually exclusive, P(A ∩ B)= P(A)P(B)

Re: Conditional Probability problem

Quote:

Originally Posted by

**mike2208** Let S be a sample space, with A is a subset S and B is a subset S. If P(A) = .6 what can be said about, P(A ∩ B),

when

(a) A and B are mutully exclusive?

(b) A is a subset B

(c) B is a subset A

(d) A′ is a subset B′

(e) A is a subset B′

Not entirely sure how to approach this problem...let me know if the first is on the right track.

a) If A and B are mutually exclusive, P(A ∩ B)= P(A)P(B)

No, $\displaystyle \mathcal{P}(A\cap B)=0$.

Mutually exclusive means $\displaystyle A\cap B=\emptyset.$

Just as $\displaystyle A\subseteq B$ means $\displaystyle A\cap B= A$

Re: Conditional Probability problem

That's right, if they are mutually exclusive then the probability they intersect is zero.

So would B subset of A mean B ∩ A = B?

Re: Conditional Probability problem

Quote:

Originally Posted by

**mike2208** That's right, if they are mutually exclusive then the probability they intersect is zero.

So would B subset of A mean B ∩ A = B?

Yes.

**One cannot do probability if one does not know all basic set operations.**

Re: Conditional Probability problem

Ok, I have all but the last.

If A' is subset of B', then B is subset of A, so P(A ∩ B) = P(B).

However, I still have not found any rules on the final question, A is subset of B'.

Re: Conditional Probability problem

Quote:

Originally Posted by

**mike2208** I still have not found any rules on the final question, A is subset of B'.

$\displaystyle A \subseteq B'\; \iff \;A \cap B = \emptyset $

Re: Conditional Probability problem

Oh thanks. I haven't seen that one before. I'm looking for the proof online. It's not listed in my textbook.

I can see that easily from a diagram, but I'm trying to find a list of important set theory laws and indentities.

Re: Conditional Probability problem

Quote:

Originally Posted by

**mike2208** Oh thanks. I haven't seen that one before. I'm looking for the proof online. It's not listed in my textbook.

I can see that easily from a diagram, but I'm trying to find a list of important set theory laws and indentities.

Think about it.

$\displaystyle A\subseteq B'$ says "all A's are not B's".