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Intro statistics question: probability of intersection

  1. Oct 5, 2017 #1
    1. The problem statement, all variables and given/known data
    If event A equals event B, then the probability of their intersection is 1. True or False?

    Apparently the correct answer is False.

    3. The attempt at a solution

    If A=B then they should overlap entirely and their intersection should be 1? The only way I see this working is if A is a subset of B and therefore they do not overlap completely, but when the question states "equals" I would think this means they are the same.
     
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  3. Oct 5, 2017 #2

    Ray Vickson

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    Suppose that in one toss of a coin we have A ={get heads} and B = {do not get tails}. Do you agree that A=B? What is their intersection? Why do you think that their intersection is 100% certain?
     
  4. Oct 5, 2017 #3
    Yes, I agree that A = B in this sense because there are two options, heads or tails - getting a head and not getting a tail is the same thing. By definition the intersection would be those elements that are in common between the events, in this case getting heads. It is 100% certain in terms of probability because the elements of each event (getting heads or not getting tails) are the same and thus their P(intersection) = 1. So isn't the statement true?
     
  5. Oct 5, 2017 #4

    Ray Vickson

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    So, I can be 100% sure to get a "head" in a coin-toss just by describing it in two ways? If I say "get heads", that has probability 1/2, and if I say "do not get tails" that has probability 1/2 also, but if I say it in two different ways it suddenly has probability 1?
     
  6. Oct 6, 2017 #5
    But it is asking about the probability of their "Intersection" so shouldn't the intersection between the two overlap completely and be 1?
     
  7. Oct 6, 2017 #6

    Ray Vickson

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    That is why I asked you to tell me what is the intersection of the events A={get heads} and B = {do not get tails}. The intersection ##A \cap B## is some subset of the sample space ##S = \{ H,T \}.## What IS that subset? Do not tell me in words; actually display the subset.
     
  8. Oct 6, 2017 #7
    Their intersection would be a subset of S say, ##Sub = \{ H \}.##
     
  9. Oct 6, 2017 #8
    ok i got it thank you
     
  10. Oct 6, 2017 #9

    Merlin3189

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    For a pack of cards;

    say, event A is getting a club, event B is getting a ten, What is the probability of their intersection?
    In this case A≠B and P(A) ≠ P(B) ≠ P(A∩B)≠P(A) , P(A)=1/4 ,P(B) = 1/13 and P(A∩B) = 1/52

    Now say event A is getting a red card and event B is getting a heart or a diamond. Then P(A) = P(B) = P(A∩B) = 1/2, since A, B and A∩B are all the same

    May be you're thinking about the logical expression ( P(A)=P(B) ) which would be true if A=B, since P(A) would = P(B).
    True and false are sometimes represented as 1 and 0.
     
  11. Oct 6, 2017 #10

    Ray Vickson

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    Right.

    Of course, if ##A = B## then ##A \cap B = A##, so ##P(A \cap B) = P(A)##.

    You may have been accidentally thinking of conditional probabilities, because for them it is true that
    $$ A = B \; \; \Rightarrow \;\; P(A | B) = 1.$$
     
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