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Help with probability question

  1. Jul 30, 2016 #1
    1. The problem statement, all variables and given/known data
    (a) Suppose a fair six-sided die is rolled once. Let A be the event that an even face occurs and B be the event that a face less than 4 occurs. Are the events A and B independent? Show this mathematically.

    (b) A fair coin is tossed three times. Let A be the event that the first toss is a Head and let B be the event all three are the same. Are the events A and B independent? Show this mathematically

    3. The attempt at a solution

    a)
    I know that:
    P(A) = 1/2
    P(B) = 1/2
    but I dont know how to show that they are independent

    b)
    I know that P(A)=1/2 , P(B) = 1/8
    but dont know how to show that they are independent

    any help would be much appreciated
     
  2. jcsd
  3. Jul 30, 2016 #2

    FactChecker

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    You omitted the "Relevant Equations" section. What is your definition of independent? It should be an equation that you can either prove to be true or false in each part of the question.
     
  4. Jul 30, 2016 #3
    p(AuB)=P(A) x P(B)

    p(AuB)=1/4

    1/4=1/2 x 1/2
    1/4=1/4

    therefore they are independent

    Would this now be correct?
     
  5. Jul 31, 2016 #4

    haruspex

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    I assume you mean that a test for independence is that P(A&B)=P(A)P(B). AuB would mean 'or'.
    Are you saying that the probability of rolling an even number less than 4 is 1/4?
     
  6. Jul 31, 2016 #5
    Sorry should say AnB=P(A) x P(B) and that P(AnB)= 1/4
     
  7. Jul 31, 2016 #6

    haruspex

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    You did not answer my question:
     
  8. Jul 31, 2016 #7
    no I am saying that it is 1/2
    P(A) = 1/2
     
  9. Jul 31, 2016 #8

    haruspex

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    No, event A is rolling an even number. A&B is rolling an even number less than four. What is the probability of that?
     
  10. Jul 31, 2016 #9
    1/6
     
  11. Jul 31, 2016 #10

    haruspex

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    Right. So is P(A&B)=P(A)P(B)?
     
  12. Jul 31, 2016 #11
    this means that they are not equal which means they are not independent

    does this mean for part b) that P(A&B)=1/8
     
  13. Jul 31, 2016 #12

    haruspex

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    Yes, and yes.
     
  14. Jul 31, 2016 #13
    so this would mean that part b would be independent
     
  15. Jul 31, 2016 #14

    FactChecker

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    Checking the details of the part b answer, I think your original post has a wrong value for P(B).
     
  16. Jul 31, 2016 #15

    haruspex

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    Right.
     
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