To find the probability of randomly selecting a person who will vote for Candidate 1 or Candidate 2, we can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
In this specific case, since Candidates 1 and 2 cannot be supported by the same voter, they are mutually exclusive. Therefore, P(A and B) = 0.
- Number of supporters for Candidate 1: 250
- Number of supporters for Candidate 2: 1,250
Now we calculate the total number of supporters for Candidates 1 and 2:
\[ \text{Total supporters for Candidate 1 or 2} = 250 + 1250 = 1500 \]
The total number of surveyed voters is 2,500.
Now, we can calculate the probability:
\[ P(\text{Candidate 1 or 2}) = \frac{\text{Total supporters for Candidate 1 or 2}}{\text{Total surveyed voters}} = \frac{1500}{2500} \]
So the answer is:
1500 over 2500.
Part 2
If an additional 1,000 voters are surveyed, the new total number of surveyed voters will be:
\[ \text{New total} = 2500 + 1000 = 3500 \]
To estimate how many voters would likely prefer Candidate 3 or Candidate 4, we can look at the survey results we have.
- Number of supporters for Candidate 3: 450
- Number of supporters for Candidate 4: 550
Adding these gives:
\[ \text{Total supporters for Candidate 3 or 4} = 450 + 550 = 1000 \]
Now, we calculate the proportion of voters who supported Candidates 3 and 4 based on the initial survey of 2,500 voters:
\[ \text{Proportion of Candidate 3 or 4} = \frac{1000}{2500} = 0.4 \]
Now, we can apply this proportion to the new total number of voters (3,500) to estimate how many would prefer either Candidate 3 or Candidate 4:
\[ \text{Estimated supporters for Candidate 3 or 4} = 0.4 \times 3500 = 1400 \]
Thus, if 1,000 more voters are surveyed, we can estimate that approximately 1,400 more voters will likely prefer Candidate 3 or 4.