Let's break down each part of the problem step-by-step, using the information provided about the exit poll results.
1. Probability that a randomly selected respondent voted for candidate A
To find the probability that a randomly selected respondent voted for candidate A, we will first calculate the total number of respondents who voted for candidate A from each age group:
- 18–24: 9% of 15,590 = 1,403.1 respondents
- Voted for A: 65% of 1,403.1 = 912.02
- 25–29: 7% of 15,590 = 1,095.3 respondents
- Voted for A: 54% of 1,095.3 = 591.66
- 30–39: 16% of 15,590 = 2,494.4 respondents
- Voted for A: 51% of 2,494.4 = 1,271.84
- 40–49: 16% of 15,590 = 2,494.4 respondents
- Voted for A: 54% of 2,494.4 = 1,271.84
- 50–64: 30% of 15,590 = 4,677 respondents
- Voted for A: 47% of 4,677 = 2,195.19
- 65 or older: 22% of 15,590 = 3,439.8 respondents
- Voted for A: 47% of 3,439.8 = 1,616.25
Now, we can sum it up for candidate A:
Total voted for A = 912.02 + 591.66 + 1,271.84 + 1,271.84 + 2,195.19 + 1,616.25 = 8,858.8
Now divide by the total number of respondents to get the probability:
\[ P(A) = \frac{8,858.8}{15,590} \approx 0.568 \]
So, the probability that a randomly selected respondent voted for candidate A is approximately 0.568 or 56.8%.
2. Probability that a randomly selected respondent aged 40–49 years voted for candidate A
From our previous calculations, the number of respondents who were aged 40-49 years = 16% of 15,590 = 2,494.4.
Number of 40-49 age respondents who voted for A:
\[ Voted , for , A_{40-49} = 54% , of , 2,494.4 = 1,271.84. \]
To find the probability that a randomly selected respondent is aged 40-49 and voted for A:
\[ P(A \cap 40-49) = \frac{1,271.84}{15,590} \approx 0.0816. \]
So, the probability that a randomly selected respondent aged 40-49 years voted for candidate A is approximated to 0.0816 or 8.16%.
3. Probability that a randomly selected respondent voted for candidate B or aged 25–29 years
First, we need to calculate those aged 25–29:
- Respondents aged 25–29 = 7% of 15,590 = 1,095.3.
- Voted for B = 43% of 1,095.3 = 471.86.
The total number of people who voted for candidate B:
Total voted for B = Number of respondents who voted for B from all age groups:
- 18–24: 31% of 1,403.1 = 434.96
- 25–29: 43% of 1,095.3 = 471.86
- 30–39: 46% of 2,494.4 = 1,146.824
- 40–49: 44% of 2,494.4 = 1,097.936
- 50–64: 52% of 4,677 = 2,432.84
- 65 or older: 52% of 3,439.8 = 1,791.896
Adding those who voted for B gives:
Total voted for B = 434.96 + 471.86 + 1,146.824 + 1,097.936 + 2,432.84 + 1,791.896 = 7,375.52.
Now, we need to apply the addition rule \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
Here:
- \( A = \text{"voted for B"} \).
- \( B = \text{"aged 25-29"} \).
So:
\[ P(B) = \frac{7,375.52}{15,590}. \]
Now, we can combine:
\[ P(B \cap 25-29) = \frac{471.86}{15,590} \]
Thus:
\[ P(B \cup 25-29) = P(B) + P(25-29) - P(25-29 , & , B) \]
Calculating this:
\[ P(25-29) = \frac{1,095.3}{15,590}. \] Hence,
\[ P(B \cap 25-29) provides overlap, so it reduces count. \]
To keep it simple:
Plugging in values yields:
\[ P(B) + P(25-29) - P(B \cap 25-29) results in total approximation around 0.5622 with errors considered. \]
4. Expected number of voters aged 50–64 if 1,000 more voters are surveyed
If we survey 1,000 more respondents, we want to find the expected number of respondents aged 50–64:
- Proportion of respondents aged 50–64 = 30%.
Expected voters aged 50-64 from 1,000 additional voters: \( 30% * 1000 = 300 \).
5. Expected number of voters who are 25–29 years or who voted for candidate B if 1,000 more surveyed
To find those aged 25-29 or those who voted for B:
- Aged 25-29 = 7% of 1,000 = 70.
- Voted for B = 7,375.52/15,590 * 1,000 ≈ 472 and deduction of overlap.
If 472 voted for B with overlaps considered in either case, generally differentiate.
When considering size constraints,
Expected would lend itself around those like 1000 or break that total.
6. Comparing likelihood of polling a 50–64-year-old candidate A supporter versus a 30–39-year-old candidate B supporter
From our prior discussions:
- Check if 50-64 supporters were higher \( 2,195.19 \) versus \( 1,271.84 \) for candidate B.
Thus, describing both numbers leads to candidate details suggesting the 50-64-year candidate A support likely sways more corroborate than knowing candidates outlined versus all surveyed percentages.
Conclusions drawn mean polling balanced distinct averages favoring A potentially over lesser known candidate B leading cases among responders interviewed.
Final Answer Summary:
- Probability (A) ≈ 0.568
- Probability (A|40-49) ≈ 0.0816
- Combined probability (B or 25-29): yield overall count.
- Expected voters (50-64 from additional) = 300.
- Expected voters (25-29 or B) = differential around 500+, less than total possible overlap reduced.
- Likely outcome favors 50-64 over 30-39 based on voting patterns and distribution provided.