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A news network published the results of an exit poll conducted during a presidential election. The survey reported 15,590 total respondents. Respondents are categorized according to age-groups. Table 4 reflects the percentage distribution of respondents and the percentage of respondents from each group who voted for candidate A or candidate B.

The entries in the first column mean that 65% of the respondents who were aged 18-24 years at that time voted for candidate A and 31% chose candidate B. This led to a 4% unreported response, which could either mean that they did not cast a vote for president or they did not want to share their votes.

Table 4. Exit Poll Survey Results
Age (in years)
18–24

9%

25–29

7%

30–39

16%

40–49

16%

50–64

30%

65 or older

22%

Voted for candidate A 65%
54%
51%
54%
47%
47%
Voted for candidate B 31%
43%
46%
44%
52%
52%
Use the results of the exit poll to answer the succeeding questions.

If a respondent is randomly selected from the database, what is the probability that that person is among those who voted for candidate A?
If a respondent is randomly selected from the database, what is the probability that that person is among those who voted for candidate A and aged 40–49 years?
If a respondent is randomly selected from the database, what is the probability that that person is among those who voted for candidate B or aged 25–29 years?
If 1,000 more voters were surveyed, what would have been the expected number of voters from this group who were 50–64 years?
If 1,000 more voters were surveyed, what would have been the expected number of voters from this group who were 25–29 years or who voted for candidate B?
If 1,000 more voters were surveyed, would it have been more likely to poll a 50–64-year-old candidate A supporter than a 30–39-year-old candidate B supporter? How can you tell?

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Answer 1

First, we need to calculate the total number of respondents who voted for candidate A based on the percentages provided in Table 4.

Total number of respondents who voted for candidate A = (65% of 9%) + (54% of 7%) + (51% of 16%) + (54% of 16%) + (47% of 30%) + (47% of 22%)
= (0.65 * 0.09 * 15590) + (0.54 * 0.07 * 15590) + (0.51 * 0.16 * 15590) + (0.54 * 0.16 * 15590) + (0.47 * 0.30 * 15590) + (0.47 * 0.22 * 15590)
= 8579.215

Therefore, the probability that a randomly selected respondent voted for candidate A is 8579.215/15590 = 0.55.

Next, we need to calculate the total number of respondents who voted for candidate A and are aged 40-49 years.

Total number of respondents aged 40-49 who voted for candidate A = 54% of 16% of 15590
= 1343.52

Therefore, the probability that a randomly selected respondent is aged 40-49 and voted for candidate A is 1343.52/15590 = 0.086.

Now, we need to calculate the total number of respondents who are aged 25-29 or voted for candidate B.

Total number of respondents aged 25-29 who voted for candidate B = 46% of 7% of 15590
= 509.23

Therefore, the probability that a randomly selected respondent is aged 25-29 or voted for candidate B is (509.23 + 1084)/15590 = 0.095.

To calculate the expected number of voters aged 50-64 if 1,000 more voters were surveyed:

Expected number of voters aged 50-64 = 0.47 * 0.30 * (15590+1000) = 2676.7

To calculate the expected number of voters aged 25-29 or who voted for candidate B if 1,000 more voters were surveyed:

Expected number of voters aged 25-29 or who voted for candidate B = (43% of 7% + 46% of 7% + 44% of 16% + 52% of 16% + 52% of 30% + 52% of 22%) * (15590+1000) = 4122.66

Finally, to determine if it would have been more likely to poll a 50-64-year-old candidate A supporter than a 30-39-year-old candidate B supporter if 1,000 more voters were surveyed, compare the probabilities of each. Based on the calculations above, the probability of polling a 50-64-year-old candidate A supporter is 0.47 * 0.30 / (0.47 * 0.30 + 0.54 * 0.16) = 0.757. The probability of polling a 30-39-year-old candidate B supporter is 0.54 * 0.16 / (0.47 * 0.30 + 0.54 * 0.16) = 0.242. Therefore, it would have been more likely to poll a 50-64-year-old candidate A supporter.