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Find a counterexample for a false statment about independent events.

  1. Oct 13, 2011 #1
    "Construct a sample space to show that the truth of this statment P(A[itex]\bigcap[/itex]B[itex]\bigcap[/itex]C)=P(A)*P(B)*P(C) is not enough for the events A,B,C to be mutually independent.

    Hint: Try finite sample spaces with equally likely simple events."

    So, my though is that I need to find a sample space with 3 events A, B, C, that are not independent, yet P(A[itex]\bigcap[/itex]B[itex]\bigcap[/itex]C)=P(A)*P(B)*P(C) is true for them.
    But I have already tried a couple of simple things and I can't seem to find a proper one. My problem propably lies in the events I take, not the spaces. I tried to use 2 tosses of fair, 6 sided dice, but my events were independent, 10 cards with the numbers 1-10 on them, 2 draws, and the card does not return to the deck. That one had dependent events, but the above statment was not true, and lastly, 2 coin tosses, but again, dependent events, untrue statment.

    Any ideas?
     
  2. jcsd
  3. Oct 13, 2011 #2
    Well, the easiest way to do this would be to let one of the events be the empty set -- then the equation P(A∩B∩C)=P(A)*P(B)*P(C) is automatically true.
     
    Last edited: Oct 13, 2011
  4. Oct 14, 2011 #3

    mathman

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    Useful example. Whole space has numbers 1 through 8, each with probability = 1/8.
    Let A = B = {1,2,3,4}
    Let C = {1,5,6,7}
    P(A)=P(B)=P(C)=1/2
    P(A and B and C)=P({1})=1/8.

    However A and B are obviously not independent. Also A and C are not independent.
     
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