Please explain steps. In a recent senatorial election, 51% of the voters in a certain district were registered as Democrats, 35% were registered as Republicans, and 14% were registered as Independents. The incumbent Democratic senator was reelected over her Republican and Independent opponents. Exit polls indicated that she gained 84% of the Democratic vote, 20% of the Republican vote, and 30% of the Independent vote. Assuming that the exit poll is accurate, what is the probability that a vote for the incumbent was cast by a registered Republican? (Round your answer to three decimal places.)

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We are given that 51% of voters registered as Democrats, 35% as Republicans, and 14% as Independents.

The Democratic candidate (the incumbent) won the election. Exit polls indicate that she got the support of 84% of Democrats, 20% of Republicans, and 30% of Independents.

Assuming the exit polls hold for the entire electorate, we are asked to find the probability that a vote for the incumbent was cast by a registered Republican.

I. This is a conditional probability question. If R is the probability that a voter is registered as a Republican and i is the probability of voting for the incumbent, then we are looking for P(R|i), the probability of R( the voter is a registered Republican) given that i (the voter voted for the incumbent.)

We use the formula `P(R|i)=(P(R cap i))/(P(i)`

`P(R cap i)` is the probability that a voter is a registered Republican and that they voted for the incumbent. This is .2(.35)=0.07 (20% of the 35% of the voters who were Republicans.)

`P(i)` is the probability that a randomly chosen voter voted for the incumbent. We have .84(.51)+.2(.35)+.3(.14)=.5404 (The voter was a Democrat or a Republican or an Independent and these are mutually exclusive.)

Thus `P(R|i)=.07/.5404~~.1295336788=25/193`

Thus the probability that a randomly chosen voter who voted for the incumbent was a registered Republican is about .130 or 13%.

II. As a check, assume there were 10000 voters. Then 5100 were registered Democrats, 3500 Republicans, and 1400 Independents. The incumbent won 4284 Democratic votes (84% of 5100), 700 Republican votes, and 420 Independent votes for a total of 5404 votes.

Of the 5404 votes, 700 were Republicans, so the probability of selecting a Republican from this group is 700/5404=.13

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