1. In a recent senatorial election, 46% of the voters in a certain district were

Question

1. In a recent senatorial election, 46% of the voters in a certain district were registered as Democrats, 30% were registered as Republicans, and 24% were registered as Independents. The incumbent Democratic senator was reelected over her Republican and Independent opponents. Exit polls indicated that she gained 80% of the Democratic vote, 24% of the Republican vote, and 34% 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.)

Explanation / Answer

Here ,we are given that:

P( democrat ) = 0.46,
P( republican ) = 0.3
P( independent ) = 0.24

Also, we are given that:

P( vote | democrat ) = 0.8,
P( vote | republican ) = 0.24,
P( vote | independent ) = 0.34

Using law of total probability, we get:

P( vote ) = P( vote | democrat )P( democrat ) + P( vote | republican )P( republican ) +P( vote | independent )P( independent )

P( vote ) = 0.8*0.46 + 0.24*0.3 + 0.34*0.24 = 0.5216

Now using bayes theores we get:

P( republican | vote ) = P( vote | republican )P( republican ) / P( vote )

P( republican | vote ) = 0.24*0.3 / 0.5216 = 0.1380

Therefore 0.1380 is the required probability here.

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