In a survey of 1000 eligible voters selected at random, it was found that 200 had a college degree. Additionally, it was found that 90% of those who had a college degree voted in the last presidential election, whereas 47% of the people who did not have a college degree voted in the last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random will have the following characteristics. (Round your answers to three decimal places.)
(a) The voter had a college degree and voted in the last presidential election.
(b) The voter did not have a college degree and did not vote in the last presidential election.
(c) The voter voted in the last presidential election.
(d) The voter did not vote in the last presidential election.
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To solve this problem, we can use the information given and set up a probability table. Let's go through each part step by step:
(a) The voter had a college degree and voted in the last presidential election:
We know that 90% of those with a college degree voted in the last presidential election. So, we can calculate this probability as:
P(college degree & voted) = P(college degree) * P(voted | college degree)
= (number of people with college degree / total eligible voters) * (percentage of college degree voters)
= (200 / 1000) * (0.9)
= 0.2 * 0.9
= 0.18
Therefore, the probability is 0.18.
(b) The voter did not have a college degree and did not vote in the last presidential election:
We know that 47% of those without a college degree voted in the last presidential election. So, we can calculate this probability as:
P(no college degree & did not vote) = P(no college degree) * P(did not vote | no college degree)
= (number of people without college degree / total eligible voters) * (percentage of no college degree voters)
= (800 / 1000) * (0.53)
= 0.8 * 0.53
= 0.424
Therefore, the probability is 0.424.
(c) The voter voted in the last presidential election:
To find this probability, we need to consider both those with college degrees who voted and those without college degrees who voted. So, we can calculate this probability as:
P(voted) = P(college degree & voted) + P(no college degree & voted)
= (number of people with college degree / total eligible voters) * (percentage of college degree voters) + (number of people without college degree / total eligible voters) * (percentage of no college degree voters)
= (200 / 1000) * (0.9) + (800 / 1000) * (0.47)
= 0.18 + 0.376
= 0.556
Therefore, the probability is 0.556.
(d) The voter did not vote in the last presidential election:
To find this probability, we need to consider both those with college degrees who did not vote and those without college degrees who did not vote. So, we can calculate this probability as:
P(did not vote) = P(college degree & did not vote) + P(no college degree & did not vote)
= (number of people with college degree / total eligible voters) * (percentage of college degree non-voters) + (number of people without college degree / total eligible voters) * (percentage of no college degree non-voters)
= (200 / 1000) * (1 - 0.9) + (800 / 1000) * (1 - 0.47)
= 0.02 + 0.424
= 0.444
Therefore, the probability is 0.444.
So, the final probabilities are:
(a) The voter had a college degree and voted in the last presidential election: 0.18
(b) The voter did not have a college degree and did not vote in the last presidential election: 0.424
(c) The voter voted in the last presidential election: 0.556
(d) The voter did not vote in the last presidential election: 0.444
To answer these questions, we need to use the given information and apply basic probability principles. Let's calculate the probabilities for each scenario.
(a) The voter had a college degree and voted in the last presidential election:
We know that 200 out of 1000 eligible voters had a college degree. Therefore, the probability of randomly selecting a voter with a college degree is 200/1000 = 0.2. Among those with a college degree, 90% voted in the last presidential election. So, the probability we are looking for is 0.2 * 0.9 = 0.18.
(b) The voter did not have a college degree and did not vote in the last presidential election:
Out of the 1000 eligible voters, 200 had a college degree. Thus, 1000 - 200 = 800 did not have a college degree. Among those without a college degree, 47% did not vote in the last presidential election. So, the probability we are looking for is 0.47 * 0.8= 0.376.
(c) The voter voted in the last presidential election:
To find this probability, we need to consider the total number of voters who voted in the last presidential election. Among those with a college degree, 90% voted, which is 0.9 * 200 = 180 voters. Among those without a college degree, 47% voted, which is 0.47 * 800 = 376 voters. Therefore, in total, 180 + 376 = 556 voters voted in the last presidential election. The probability we are interested in is 556/1000 = 0.556.
(d) The voter did not vote in the last presidential election:
To calculate this probability, we complement the previous probability. The probability that a voter did not vote in the last presidential election is 1 - 0.556 = 0.444.
To summarize:
(a) The probability that a voter had a college degree and voted in the last presidential election is 0.18.
(b) The probability that a voter did not have a college degree and did not vote in the last presidential election is 0.376.
(c) The probability that a voter voted in the last presidential election is 0.556.
(d) The probability that a voter did not vote in the last presidential election is 0.444.