A poll is given, showing 55% are in favor of a new building project.

If 5 people are chosen at random, what is the probability that exactly 4 of them favor the new building project?


To find the probability that exactly 4 out of 5 randomly chosen people favor the new building project, we can use the binomial probability formula:

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

Where:
- \( n \) is the total number of trials (in this case, 5),
- \( k \) is the number of successes (in this case, 4),
- \( p \) is the probability of success (in this case, 0.55),
- \( \binom{n}{k} \) is the binomial coefficient.

First, we calculate the binomial coefficient \( \binom{5}{4} \):

\[
\binom{5}{4} = \frac{5!}{4!(5-4)!} = 5
\]

Next, we calculate \( p^k \) and \( (1-p)^{n-k} \):

\[
p^k = (0.55)^4 \approx 0.09150625
\]
\[
(1-p)^{n-k} = (0.45)^1 = 0.45
\]

Now, we can plug these values into the binomial probability formula:

\[
P(X = 4) = \binom{5}{4} \cdot (0.55)^4 \cdot (0.45)^1
\]

\[
P(X = 4) = 5 \cdot 0.09150625 \cdot 0.45 \approx 0.2059
\]

Thus, the probability that exactly 4 out of 5 people favor the new building project is approximately **0.2059**, or **20.59%**.

### Questions:
1. What if we wanted to calculate the probability of exactly 3 people favoring the project?
2. How would the probability change if 70% were in favor instead?
3. What is the probability that at least 4 people favor the project?
4. How would you explain the binomial probability concept to someone new to statistics?
5. What are other applications of binomial probability in real-life scenarios?

### Tip:
Always verify the assumptions of a binomial distribution: the number of trials should be fixed, each trial should be independent, and the probability of success should remain constant throughout the trials.

A poll is given, showing 55% are in favor of a new building project. If 5 people are chosen at random, what is the probability that exactly 4 of them favor the new building project?

Explore how to calculate the probability that exactly 4 out of 5 randomly selected people favor a new building project with a support rate of 55%. This page covers the binomial probability formula and key concepts in probability theory.

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