A poll is given, showing 60% are in favor of a new building project in Chicago. If 40 people are chosen at random, answer the following. Round your final answers to 4 decimal places.
a. What is the probability that exactly 20 of them favor the new building project?
.0554
b. What is the probability that less than 20 of them favor the new building project?
c. What is the probability that more than 20 of them favor the new building project?
.8702
d. What is the probability that exactly 23 of them favor the new building project?
e. What is the probability that at least 23 of them favor the new building project?
f. What is the probability that at most 23 of them favor the new building project?

We are dealing with a binomial distribution problem since the experiment consists of \( n = 40 \) trials (people) with two possible outcomes: favoring or not favoring the new building project. The probability that an individual favors the project is \( p = 0.60 \), and the probability that an individual does not favor the project is \( 1 - p = 0.40 \).

The binomial probability formula is given by:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
where:
- \( X \) is the random variable representing the number of people favoring the project,
- \( k \) is the specific number of people we are interested in,
- \( n \) is the total number of trials (40 people),
- \( p \) is the probability of success on a single trial (favoring the project).

Let’s solve each part step by step.

### a. Probability that exactly 20 people favor the project:
We are given this result as \( P(X = 20) = 0.0554 \).

### b. Probability that less than 20 people favor the project:
We need to find \( P(X < 20) \), which is the sum of probabilities from \( X = 0 \) to \( X = 19 \).

\[
P(X < 20) = P(X = 0) + P(X = 1) + \dots + P(X = 19)
\]

### c. Probability that more than 20 people favor the project:
We are given this result as \( P(X > 20) = 0.8702 \).

### d. Probability that exactly 23 people favor the project:
We can use the binomial formula for \( X = 23 \).

\[
P(X = 23) = \binom{40}{23} (0.60)^{23} (0.40)^{17}
\]

### e. Probability that at least 23 people favor the project:
This is \( P(X \geq 23) \), which can be calculated as:
\[
P(X \geq 23) = P(X = 23) + P(X = 24) + \dots + P(X = 40)
\]
Alternatively, we can use:
\[
P(X \geq 23) = 1 - P(X \leq 22)
\]

### f. Probability that at most 23 people favor the project:
This is \( P(X \leq 23) \), which is the sum of probabilities from \( X = 0 \) to \( X = 23 \):
\[
P(X \leq 23) = P(X = 0) + P(X = 1) + \dots + P(X = 23)
\]

Let me now calculate the values for parts b, d, e, and f.Here are the results for the remaining parts of the problem:

- **b. Probability that less than 20 people favor the project:** \( P(X < 20) = 0.0744 \)
- **d. Probability that exactly 23 people favor the project:** \( P(X = 23) = 0.1204 \)
- **e. Probability that at least 23 people favor the project:** \( P(X \geq 23) = 0.6885 \)
- **f. Probability that at most 23 people favor the project:** \( P(X \leq 23) = 0.4319 \)

If you have any questions or need further clarification, feel free to ask!

Here are some related questions to expand your understanding:
1. What is the expected number of people who favor the project in this scenario?
2. What is the variance of the number of people who favor the project?
3. What is the probability that exactly 25 people favor the project?
4. How does the probability change if the sample size increases to 50 people?
5. What is the probability that fewer than 10 people favor the project?

**Tip:** The binomial distribution is a great model when you have independent trials with only two possible outcomes (success/failure) in each trial!

A poll is given, showing 60% are in favor of a new building project in Chicago. If 40 people are chosen at random, answer the following. Round your final answers to 4 decimal places. a. What is the probability that exactly 20 of them favor the new building project? b. What is the probability that less than 20 of them favor the new building project? c. What is the probability that more than 20 of them favor the new building project? d. What is the probability that exactly 23 of them favor the new building project? e. What is the probability that at least 23 of them favor the new building project? f. What is the probability that at most 23 of them favor the new building project?

Explore the binomial distribution in a real-world context with this probability problem. Given that 60% are in favor of a building project, and a random sample of 40 people is chosen, calculate various probabilities including exact, at least, and at most values. Detailed step-by-step solutions provided.

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