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Math Help - Bayes rule solution check

  1. #1
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    Bayes rule solution check

    A population of voters contains 40% Republicans and 60% Democrats. It is reported that 30% of the Republicans and 70% of the Democrats favor an election issue. A person chosen at random from this population is found to favor the issue in question. Find the conditional probability that this person is a Democrat.

    Here is what I did.

    Let P(r) person chosen is a republicant = 0.4
    Let P(d) person chosen is a democrat = 0.6
    P(a|r) person supports issue given he\she is republicant = 0.3
    P(a|d) person supports issue given he\she is democrat = 0.7

    Event of interest = P(d|a) (person is a democrat given he\she supports the issue)

    Bayes' formula P(d|a) = P(a and d) / P(a)

    where I found p(a)= 0.3*0.4 + 0.7*0.6 = 0.54

    hence P(d|a) = 0.7*0.6 / 0.54 = 0.7777 which is about 78%

    Does this sound like a right way to solve this problem or I am doing something wrong here?

    Thank You
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  2. #2
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    Quote Originally Posted by somestudent2 View Post
    A population of voters contains 40% Republicans and 60% Democrats. It is reported that 30% of the Republicans and 70% of the Democrats favor an election issue. A person chosen at random from this population is found to favor the issue in question. Find the conditional probability that this person is a Democrat.

    Here is what I did.

    Let P(r) person chosen is a republicant = 0.4
    Let P(d) person chosen is a democrat = 0.6
    P(a|r) person supports issue given he\she is republicant = 0.3
    P(a|d) person supports issue given he\she is democrat = 0.7
    Let 'e' stand for favoring the election issue.

    Then (your question sound a little ambigous to me, but I think you mean this),
    P(d|a) = \frac{P(a|d)P(d)}{P(a|r)P(r)+P(a|d)P(d)}
    Now fill in those values.
    P(d) = P(r) = 1/2.
    While, P(a|r) = .3 and P(a|d) = .7
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  3. #3
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    Hi, thanks for your quick response. I have a question. How did you get P(d)=P(r) = 1/2. I thought we are given that there are 40% Republicans so P(r)=0.4 and 60% Democrats so P(d)=0.6 ?

    Thanks again.
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  4. #4
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    Quote Originally Posted by somestudent2 View Post
    Hi, thanks for your quick response. I have a question. How did you get P(d)=P(r) = 1/2. I thought we are given that there are 40% Republicans so P(r)=0.4 and 60% Democrats so P(d)=0.6 ?
    You are correct. The other is mistake made in haste I am sure.
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  5. #5
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    Quote Originally Posted by somestudent2 View Post
    Hi, thanks for your quick response. I have a question. How did you get P(d)=P(r) = 1/2. I thought we are given that there are 40% Republicans so P(r)=0.4 and 60% Democrats so P(d)=0.6 ?

    Thanks again.
    Yes I made a mistake. When you said "at random" I was thinking 50/50 type random. But the problem says otherwise.
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