In a population where 85%
 of voters prefer Candidate A, an organization conducts a poll of 
 20 voters. Find the probability that 18 of the 20 voters will prefer Candidate A.

(Report answer accurate to at least 4 decimal places.)


The situation described is a binomial probability problem. The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( P(X = k) \) is the probability of exactly \( k \) successes in \( n \) trials.
- \( n \) is the number of trials (in this case, 20 voters).
- \( k \) is the number of successes (in this case, 18 voters preferring Candidate A).
- \( p \) is the probability of success on a single trial (in this case, 0.85, since 85% of voters prefer Candidate A).
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials.

Now let's calculate it:

\[
P(X = 18) = \binom{20}{18} (0.85)^{18} (1 - 0.85)^{2}
\]

I'll compute this step by step.The probability that exactly 18 out of 20 voters will prefer Candidate A is approximately \( 0.2293 \), or 22.93%.

Would you like a detailed explanation of how this was calculated or any clarifications?

Here are 5 related questions you might find interesting:
1. What is the probability that exactly 19 voters will prefer Candidate A?
2. How does the probability change if only 10 voters were polled?
3. What is the expected number of voters preferring Candidate A in this poll?
4. How does the probability distribution look for different numbers of voters preferring Candidate A?
5. How would the result change if the probability of preferring Candidate A was 90% instead of 85%?

**Tip:** The binomial distribution is useful when you're interested in the number of successes in a fixed number of independent trials, each with the same probability of success.

In a population where 85% of voters prefer Candidate A, an organization conducts a poll of 20 voters. Find the probability that 18 of the 20 voters will prefer Candidate A. (Report answer accurate to at least 4 decimal places.)

Explore how to calculate the probability that 18 out of 20 voters prefer Candidate A in a population where 85% favor the candidate. Using the binomial distribution formula, this detailed solution is suitable for high school students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation