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Math Help - Bayes' Theorem problem.

  1. #1
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    Bayes' Theorem problem.

    I have no idea how to solve this problem, need help badly!!

    In a given county, records show that of the registered voters, 45% are Democrats, 30% are Republicans, and 25% are Independents, In an election, 70% of the Democrats, 30% of the Republicans, and 90% of the Independents voted in favor of a parks and recreation bond proposal. If registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Republican, An Independent? A democrat?
    Last edited by mr fantastic; December 13th 2011 at 01:31 AM. Reason: Title.
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  2. #2
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    Re: Bayes' Theorem problem need help

    Quote Originally Posted by pauly215 View Post
    I have no idea how to solve this problem, need help badly!!

    In a given county, records show that of the registered voters, 45% are Democrats, 30% are Republicans, and 25% are Independents, In an election, 70% of the Democrats, 30% of the Republicans, and 90% of the Independents voted in favor of a parks and recreation bond proposal. If registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Republican, An Independent? A democrat?
    Let R,D,I,Y denote the that a voter is republican, democrat, independedent, voted yea. Now Bayes tells us that:

    P(R|Y)=\frac{P(Y|R)P(R)}{P(Y)}=\frac{P(Y|R)P(R)}{P  (Y|R)P(R)+P(Y|D)P(D)+P(Y|I)P(I)}

    Now you are told the values of all the probabilities in the expression on the right, so any further problems?
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  3. #3
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    Re: Bayes' Theorem probablility

    Hello, pauly215!

    I have no idea how to solve this problem.
    Why do you insist on saying that?
    Really, NO idea?
    Your title suggests that you MIGHT use Bayes' Theorem.


    In a given county, records show that of the registered voters, 45% are Democrats,
    30% are Republicans, and 25% are Independents.
    In an election, 70% of the Democrats, 30% of the Republicans, and 90% of the
    Independents voted in favor of a parks and recreation bond proposal.

    If a registered voter chosen at random is found to have voted in favor of the bond,
    what is the probability that the voter is a Republican? .An Independent? .A Democrat?

    We have this data: . \begin{array}{|c|c|c|} \text{Party} & \text{Proportion} & \text{voted for} \\ \hline \text{Dem.} & 45\% & 70\% \\ \text{Rep.} & 30\% & 30\% \\ \text{Ind.} & 25\% & 90\% \\ \hline \end{array}

    We have:
    . . \begin{array}{cccccccccc}P(\text{Dem}\,\wedge\text  {for}) &=& 0.45\cdot0.70 &=& 0.315 \\ P(\text{Rep}\,\wedge\text{for}) &=& 0.30\cdot0.30 &=& 0.090 \\ P(\text{Ind}\,\wedge\text{for}) &=& 025\cdot 0.90 &=& 0.225 \end{array}

    Hence: . P(\text{for}) \:=\:0.315 + 0.090 + 0.225 \:=\:0.630


    Therefore, we have:

    . . \begin{array}{cccccccccc}P(\text{Dem}\,|\,\text{fo  \!\!r}) &=& \dfrac{P(\text{Dem}\,\wedge\,\text{for})}{P(\text{  \!for})} &=& \dfrac{0.315}{0.630} &=& 0.500  \\ \\[-3mm] P(\text{Rep}\,|\,\text{for}) &=& \dfrac{P(\text{Rep}\,\wedge\,\text{for})}{P(\text{  for})} &=& \dfrac{0.090}{0.630} &\approx& 0.143 \\ \\[-3mm] P(\text{Ind}\,|\,\text{for}) &=& \dfrac{P(\text{Ind}\,\wedge\,\text{for})}{P(\text{  for})} &=&\dfrac{0.225}{0.630} &\approx& 0.357    \end{array}

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