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Math Help - A ⊂ b ⇒ p (a) ≤ p (b)

  1. #1
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    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) ?
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  2. #2
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    If A represents a subset of the events corresponding to B, then if A occurs, B occurs.
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  3. #3
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    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...
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    If you want to prove in a more formal way (instead of using the suggestion above) how can you write  A \cup B as a union of disjoint sets? And more easily, what is  A \cup B if A \subset B ?
    Because if you have X and Y disjoint, i.e. X \cap Y = \emptyset ten by the axioms of probability you know that P(X \cup Y) = P(X) + P(Y) (You will also need to use the fact that for any event E, P(E) \geq 0.) If your not used to work with sets you can try by using venn diagrams.
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  5. #5
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    sorry, when I started writing your reply wasn't here
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  6. #6
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    Quote Originally Posted by Moo View Post
    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?
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  7. #7
    Moo
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    Quote Originally Posted by icemanfan View Post
    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

    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
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  8. #8
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    Quote Originally Posted by Moo View Post
    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.
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