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Probability 4

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data

    In a certain club, one-fifth of its members are smokers. One-sixth of its male members are smokers. Among the non-smoking members, one-eighth are female. A member is randomly chosen.

    (a) What is the probability that this member is male;
    (b) If this member is female, what is the probability that she is a non-smoking member.

    (Answers:
    (a) 21/25
    (b) 5/8)

    2. Relevant equations

    Probability Formulae

    3. The attempt at a solution

    One of my friends told me the followings in order to get the final answer correct.

    However, I don't know how can he get them.

    (a)
    (1 - 1/5) * (1 - 1/8) / (1 - 1/6)
    = 7/10 / (5/6)
    = 21/25
    (b)
    (1 - 1/5) * 1/8 / (1 - 21/25)
    = 1/10 / (4/25)
    = 1/10 * 25/4
    = 5/8

    I tried to write a tree diagram as follows:

    S = 1/5
    NS = 4/5

    S.M = 1/6
    S.F = 5/6

    NS.M = 7/8
    NS.F = 1/8

    S: Smoker; NS: Non-Smoker; M: Male; F: Female

    Did my tree diagram correct?

    My attempt on part (a):
    P(M)
    = P(M|S)P(S) + P(M|NS)P(NS)
    = 1/6 * 1/5 + 7/8 * 4/5
    = 11/15

    My attempt on part (b):
    P(NS|F)
    = P(NS and F) / (P(F|S)P(S) + P(F|NS)P(NS))
    = 4/5 * 1/8 / (1/5 * 5/6 + 4/5 * 1/8)
    = 3/8

    But my answers are not correct.

    Can anyone tell me how to solve this question?

    Thank you very much!
     
  2. jcsd
  3. Mar 25, 2009 #2
    P(S|M) is not the same as P(M|S) this was your error on both the parts. Do you know the formula to obtain P(M|S) from P(S|M)?
     
  4. Mar 25, 2009 #3
    P(S|M)
    = P(S and M) / P(M)
    = P(M|S)P(S) / P(M)?

    Did my tree diagram correct?

    Thank you very much!
     
  5. Mar 25, 2009 #4
    I don't know about tree diagrams but one of them looks false. Try to just write it out as P(A|B). "One-sixth of its male members are smokers", what are you given?

    As an example, 2/5 of the women population read Heat magazine, so P(reading heat| it's a woman)=2/5. The information given to you is that it is a woman, restated it says given that the person is a woman, the probability of the person reading Heat is 2/5.

    Your formula is correct (Bayesian formula). Try to rethink about what the question says. I hope that example helps. If not let me know.
     
  6. Mar 26, 2009 #5
    P(NS|M)
    = P(M|NS) P(NS) / P(M)

    P(M)
    = P(M|NS) P(NS) / P(NS|M)
    = P(M|NS) P(NS) / (1 - P(S|M))
    = 7/8 x 4/5 / (1 - 1/6)
    = 21/25

    P(NS|F)
    = P(F|NS) P(NS) / P(F)
    = P(F|NS) P(NS) / (1 - P(M))
    = 1/8 x 4/5 / (1 - 21/25)
    = 5/8

    Am I right?
     
  7. Mar 26, 2009 #6
    Yep looks fine to me :approve:
     
  8. Mar 26, 2009 #7
    Thanks for your helps!!!
     
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