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Math Help - questions need answers

  1. #1
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    Lightbulb questions need answers

    20. if (A∩B∩C)≠0 and P(C|A∩B)=P(C|B) , show that : P(A|B∩C)=P(A|B)

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    21 . suppose that A,B and c are independent events .show that
    i. A and (A∩B) are independent.
    ii. A and (AUB) are independent.
    iii. A^c and (A∩c^c) are independent.

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    22. in a city ,25% of the cars emit pollutions to the air of the city .if a car failed to pass the emission of air pollutants with probability 0.99 and if a car that is not emitting air pollutants also fails the test with probability 0.17 .what is the probability that a car that fails the test actually emits air pollution ?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by compufatwa View Post
    20. if (A∩B∩C)≠0 and P(C|A∩B)=P(C|B) , show that : P(A|B∩C)=P(A|B)
    P(A|B \cap C)=\frac{P(A \cap B \cap C)}{P(B \cap C)}=\frac{P(C \cap A \cap B)}{P(B \cap C)} = \frac{P(C|A \cap B)P(A \cap B)}{P(B \cap C)}
     = \frac{P(C|B)P(A \cap B)}{P(B \cap C)} = \frac{\frac{P(C \cap B)}{P(B)}P(A \cap B)}{P(B \cap C)} = \frac{P(A \cap B)}{P(B)} = P(A|B)
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    Quote Originally Posted by compufatwa View Post
    21 . suppose that A,B and c are independent events .show that
    i. A and (A∩B) are independent.
    ii. A and (AUB) are independent.
    iii. A^c and (A∩c^c) are independent.
    To show independence, we need to show that
    P(A \cap B) = P(A)P(B)
    so,
    i) \, P(A \cap (A \cap B)) = P((A \cap A) \cap B) = P(A \cap B)
    = P(A)P(B) = P(A \cap A)P(B) = P(A)P(A)P(B) = P(A)P(A \cap B)
    im not sure with it (i.e. if A is independent to itsefl)..
    just do the same thing with the others
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