In a survey involving 550 likely Democratic voters and 600 likely Republican voters, the question "Do you support or oppose legislation that would require registration of all handguns?" was asked. The following results were obtained. If a randomly chosen respondent in the survey answered "Oppose," what is the probability that he or she is a likely Democratic voter?
A) 0.7317
B) 0.2683
C) 0.2927
D) 0.1783
E) 0.6707

Question

In a survey involving 550 likely Democratic voters and 600 likely Republican voters, the question "Do you support or oppose legislation that would require registration of all handguns?" was asked. The following results were obtained.  If a randomly chosen respondent in the survey answered "Oppose," what is the probability that he or she is a likely Democratic voter?
A) 0.7317 
B) 0.2683 
C) 0.2927 
D) 0.1783 
E) 0.6707

Quizplus · Accepted Answer

We need to use Bayes' theorem to calculate the probability that a respondent who answered "Oppose" is a likely Democratic voter. Let D be the event that a respondent is a likely Democratic voter, and O be the event that a respondent opposes the legislation.

We want to find P(D|O), the conditional probability that a respondent is a likely Democratic voter given that they oppose the legislation.

We can use Bayes' theorem:

P(D|O) = P(O|D) * P(D) / P(O)

where P(O|D) is the probability that a likely Democratic voter opposes the legislation, P(D) is the probability that a respondent is a likely Democratic voter, and P(O) is the overall probability that a respondent opposes the legislation (regardless of party affiliation).

From the table, we see that 66% of likely Democratic voters oppose the legislation, so P(O|D) = 0.66.

There are 550 likely Democratic voters out of a total of 550 + 600 = 1150 likely voters, so P(D) = 0.478.

The table shows that a total of 491 + 252 = 743 respondents oppose the legislation, so P(O) = 743/1150 = 0.646.

Now we can plug in the values:

P(D|O) = 0.66 * 0.478 / 0.646

P(D|O) = 0.4907

Therefore, the probability that a respondent who opposes the legislation is a likely Democratic voter is approximately 0.2683 (or 26.83%).

In a Survey Involving 550 Likely Democratic Voters and 600