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Probability question

  1. Mar 21, 2016 #1
    1. The problem statement, all variables and given/known data
    For the three events A, B and C, P (exactly one of the events A or B occurs) = P (exactly one of the events B or C occurs) = P (exactly one of the events C or A occurs) = p ,
    and P (all the three events occur simultaneously) = p2 ,
    where 0 < p < ½ .
    Then the probability of at least one of the three events A, B and C occurring is:

    Ans: ## \dfrac{3p+2p^2}{2} ##

    2. Relevant equations


    3. The attempt at a solution
    http://s3.amazonaws.com/minglebox-p...ata-0000-fdbffe7622c53ecd0122c5c50d0b0334.gif
    NOTE: The regions shown do not overlap with each other, i.e. P(A) ≠ region 1, instead P(A) = region ( 1 + 2 + 4 + 5).
    By symmetry, I assume region 1 = 3 = 7 = p/2
    Region 5 = p2
    Again by symmetry, region 2 = 4 = 6 = x.
    I need to find x because the probability that I got to calculate is the sum of all the regions, ie.
    ## \dfrac{3p}{2} + 3x + p^2 ##
    I don't know how. I initially though that the sum of all probabilities might be 1, but then it doesn't say that this is the case. For all we know, there might be 10 more events. So, I dismissed that option. I don't know how to proceed from here.
     
  2. jcsd
  3. Mar 21, 2016 #2

    PeroK

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    Why not just write down the equations you know in terms of the areas 1-7? You don't have to assume symmetry.
     
  4. Mar 21, 2016 #3
    How does that help?
    Here:
    1 + 3 = 3 + 7 = 7 + 1 = p
    5 = p2
    I still don't know 2, 4, 6
    Also, from the answer, I just noticed that 2, 4 and 6 must be 0. Which is very surprising.
     
  5. Mar 21, 2016 #4

    PeroK

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    Those equations are not right. Look more carefully at your diagram. What areas represent "exactly one of A or B"?
     
  6. Mar 21, 2016 #5
    Got it. 1 + 4 + 3 + 6 represents exactly one of A and B. Thank you so much :)
     
  7. Mar 21, 2016 #6
    how did you come up with that Venn's diagram? is it given or you came up with it?. If it's given, then just write down the equation of probability of the union of events.
     
  8. Mar 21, 2016 #7
    Oh no, it wasn't given. I got the answer now.
     
  9. Mar 21, 2016 #8
    My intuition tells me that you have to use negation to find AnB which corresponds to the areas 2 and 5. I don't know if that's correct
     
  10. Mar 21, 2016 #9
  11. Mar 21, 2016 #10
    Ummm I would be glad to know the way you came up with the answer because I don't know XD.
     
  12. Mar 21, 2016 #11
    Oh sure!
    1 + 4 + 3 + 6 = p
    2 + 3 + 4 + 7 = p
    1 + 2 + 7 + 6 = p
    Adding them,
    2 ( 1 + 2 + 3 + 4 + 6 + 7 ) = 3p
    (1+2+3+4+6+7) = 3p/2
    5 = p2
    Hence, 1 + 2 + 3 + 4 + 5 + 6 + 7 = 3p/2 + p2
     
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