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.