# A ⊂ b ⇒ p (a) ≤ p (b)

• Feb 16th 2010, 11:56 AM
A ⊂ b ⇒ p (a) ≤ p (b)
A ⊂ B ⇒ P (A) ≤ P (B) using result of the Axioms of Probability How can I proof
P(B) ≤ P (AUB ) and
P (AB)
≤ P(A) ?
• Feb 16th 2010, 12:06 PM
icemanfan
If A represents a subset of the events corresponding to B, then if A occurs, B occurs.
• Feb 16th 2010, 12:25 PM
Moo
Hello,

It's not a matter of events, since we want to use the axioms, what you said is not valid ^^'

To prove that A ⊂ B ⇒ P (A) ≤ P (B), just consider the disjoint sets BnA and BnA', where A' denotes the complement of A.
Their union makes B, and by the third axiom of probability, you can conclude...

P(B) ≤ P(AUB) comes from the fact that B ⊂ AUB
P (AB) ≤ P(A) comes from the fact that AB ⊂ A

That's all...
• Feb 16th 2010, 12:32 PM
johanS
If you want to prove in a more formal way (instead of using the suggestion above) how can you write\$\displaystyle A \cup B\$ as a union of disjoint sets? And more easily, what is \$\displaystyle A \cup B\$ if \$\displaystyle A \subset B\$ ?
Because if you have X and Y disjoint, i.e. \$\displaystyle X \cap Y = \emptyset\$ ten by the axioms of probability you know that \$\displaystyle P(X \cup Y) = P(X) + P(Y)\$ (You will also need to use the fact that for any event \$\displaystyle E, P(E) \geq 0\$.) If your not used to work with sets you can try by using venn diagrams.
• Feb 16th 2010, 12:34 PM
johanS
• Feb 16th 2010, 12:55 PM
icemanfan
Quote:

Originally Posted by Moo
It's not a matter of events, since we want to use the axioms, what you said is not valid ^^'

Sorry, I was trying to convey the following idea: Let A represent snowing in Chicago and B represent snowing somewhere in Illinois. Clearly, A is a subset of B and if A occurs, then B occurs. Isn't this a universal pattern?
• Feb 16th 2010, 01:01 PM
Moo
Quote:

Originally Posted by icemanfan
Sorry, I was trying to convey the following idea: Let A represent snowing in Chicago and B represent snowing somewhere in Illinois. Clearly, A is a subset of B and if A occurs, then B occurs. Isn't this a universal pattern?

Yes, but it can't be taken as a formal proof of what's being said here (Nod)

Why would A be a subset of B, though they're events ? There's a degree of abstract that can't be found if we talk about "if event A occurs, then so does B". It's a mere translation of the inequality between the probabilities that there are in the first message.

I don't know if I explained well what I thought, I hope you understand what I want to say :(
• Feb 16th 2010, 01:18 PM
icemanfan
Quote:

Originally Posted by Moo
I don't know if I explained well what I thought, I hope you understand what I want to say :(

I think I understand what you are saying. I could elaborate on my point, but you have aptly demonstrated how to solve the problem, so there really is no need for that.