# Conditional Probability problem

• Sep 19th 2011, 10:52 AM
mike2208
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)
• Sep 19th 2011, 11:25 AM
Plato
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, $\mathcal{P}(A\cap B)=0$.
Mutually exclusive means $A\cap B=\emptyset.$

Just as $A\subseteq B$ means $A\cap B= A$
• Sep 19th 2011, 11:29 AM
mike2208
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?
• Sep 19th 2011, 11:46 AM
Plato
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.
• Sep 19th 2011, 12:08 PM
mike2208
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'.
• Sep 19th 2011, 12:59 PM
Plato
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'.

$A \subseteq B'\; \iff \;A \cap B = \emptyset$
• Sep 19th 2011, 01:09 PM
mike2208
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.
• Sep 19th 2011, 01:13 PM
Plato
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.

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