In an election, 65% of the voters were Republicans, and 35% were not. Of the Rep

Question

In an election, 65% of the voters were Republicans, and 35% were not. Of the Republicans, 70% voted for Candidate X, and of the non-Republicans, 15% voted for Candidate X. Consider a randomly selected voter. Define A = Voter is Republican. B = Voted for Candidate X. (a) Find the values. (Give the answers to two decimal places.) P(A) = P(AC) = P(B|A) = P(B|AC) = (b) Find P(A and B). (Give the answer to three decimal places.) Write in words what outcome it represents. This is the probability of being a Republican and voting for Candidate X. This is the probability of being a non-Republican and voting for Republican.     This is the probability of being a Republican and voting for Republican. This is the probability of being a non-Republican and voting for Candidate X. (c) Find P(AC and B). (Give the answer to four decimal places.) Write in words what outcome it represents. This is the probability of being a non-Republican and voting for Republican. This is the probability of being a Republican and voting for Candidate X.     This is the probability of being a non-Republican and voting for Candidate X. This is the probability of being a Republican and voting for Republican. (d) Using the results in parts (b) and (c), find P(B). (Give the answer to four decimal places. Hint: The events in parts (b) and (c) cover all of the ways in which B can happen.) (e) Use the result in part (d) to state what percent of the vote Candidate X received. (Give the answer to two decimal places.) %

Explanation / Answer

Answer:

Based on the given information:

A = Voter is Republican

B = Voted for Candidate X

Percentage of Republican Voters P(A) = 65% = 0.65

Percentage of Non-Republican Voters P(AC)= 1 - 0.65 = 0.35

Percentage of Republicans who voted for Candidate X P(B/A) = 70% = 0.70

Percentage of Non-Republicans who voted for Candidate X P(B/AC) = 15% = 0.15

(a) Find the values. (Give the answers to two decimal places.)

(b) Find P(A and B). (Give the answer to three decimal places.) Write in words what outcome it represents.

P(A and B) = P(B/A)*P(A) = 0.70*0.65 = 0.455

(c) Find P(AC and B). (Give the answer to four decimal places.) Write in words what outcome it represents.

P(AC and B) = P(B/AC)*P(AC) = 0.15*0.35 = 0.0525

(d) Using the results in parts (b) and (c), find P(B). (Give the answer to four decimal places. Hint: The events in parts (b) and (c) cover all of the ways in which B can happen.)

P(B) = P(A and B) + P(AC and B) = 0.455 + 0.0525 = 0.5075

(e) Use the result in part (d) to state what percent of the vote Candidate X received. (Give the answer to two decimal places.) %

Percent of votes received by Candidate X = 50.75%

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