Bayes' Theorem problem.

**pauly215** · December 12th 2011, 06:24 PM

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?

**CaptainBlack** · December 12th 2011, 07:11 PM

Originally Posted by pauly215

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?

**Soroban** · December 12th 2011, 07:25 PM

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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