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Homework Help: Help with my 2 easy but hard probability questions!

  1. Sep 25, 2010 #1
    1.
    Suppose P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
    a) Must we have P((0,1/2]) <= 1/3 ?
    b) Must we have P([0,1/2]) <= 1/3 ?

    My answer - Please correct me if i am wrong
    a)

    P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
    lim P([0,1/2])
    n-> infinity

    so then P((0,1/2]) is a subset of P([1/n,1/2]) for all n = 1,2,3,...
    because any number in (0,1/2] can lie in [1/n,1/2] provided the correct n value is in -> 1,2,3...

    so true
    b)

    P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
    lim P([0,1/2])
    n-> infinity

    so then P([0,1/2]) does not equal P([1/n,1/2]) for all n = 1,2,3,...
    because as n approaches infinity, the limit of 1/n goes to 0, but doesn't touch 0

    so false

    2.

    Suppose P((0,1/2]) = 1/3. Prove that there is some n such that P([1/n,1/2]) > 1/4.

    lim P([1/n,1/2]) = P([0,1/2]) > 1/3 > 1/4
    n -> infinity

    i don't think this is entirely right, i sorta guessed...
     
  2. jcsd
  3. Oct 1, 2010 #2

    vela

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    Staff Emeritus
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    P((0,1/2]) and P([1/n,1/2]) are numbers. One is not a subset of the other. Also, you have the relationship backwards: [1/n,1/2]⊂(0,1/2], not (0,1/2)⊂[1/n,1/2] as you claimed.
    Right.
    It's not correct. For one thing, you don't know that P([0,1/2])>1/3.

    What kind of properties about sets and set operations as it pertains to probabilities do you know? (This would have been what you should have written under "relevant equations" in the provided template.)
     
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