In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 12% of voters are Independent. A survey asked 29 people to identify themselves as Democrat, Republican, or Independent.

A. What is the probability that none of the people are Independent?

Probability = 

B. What is the probability that fewer than 6 are Independent?

Probability = 

C. What is the probability that more than 2 people are Independent?

Probability = 

This problem involves **binomial probability** because each person surveyed either identifies as Independent (success) or not (failure), with a fixed probability of success (12%).

Given:
- The probability that a person is Independent: \( p = 0.12 \).
- The number of people surveyed: \( n = 29 \).
- The number of successes (people identifying as Independent) is denoted by \( X \).

The binomial distribution formula is:

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

where \( \binom{n}{k} \) is the binomial coefficient, \( k \) is the number of successes, \( p \) is the probability of success, and \( n \) is the total number of trials.

Let's solve each part:

### A. Probability that none of the people are Independent
This corresponds to \( P(X = 0) \).

\[
P(X = 0) = \binom{29}{0} (0.12)^0 (0.88)^{29} = (1) \cdot (1) \cdot (0.88)^{29}
\]

Calculating \( (0.88)^{29} \):

\[
P(X = 0) \approx 0.072
\]

### B. Probability that fewer than 6 are Independent
This corresponds to \( P(X < 6) \), which is the sum of probabilities for \( X = 0, 1, 2, 3, 4, 5 \).

\[
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

Each of these terms can be computed using the binomial formula. For example:

\[
P(X = 1) = \binom{29}{1} (0.12)^1 (0.88)^{28}
\]
\[
P(X = 2) = \binom{29}{2} (0.12)^2 (0.88)^{27}
\]

Summing up these probabilities gives the total:

\[
P(X < 6) \approx 0.998
\]

### C. Probability that more than 2 people are Independent
This corresponds to \( P(X > 2) \), which is the complement of \( P(X \leq 2) \).

\[
P(X > 2) = 1 - P(X \leq 2)
\]

First, we calculate \( P(X \leq 2) \):

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

Using the binomial probabilities for each term:

\[
P(X = 0) \approx 0.072, \quad P(X = 1) \approx 0.283, \quad P(X = 2) \approx 0.396
\]

Summing these:

\[
P(X \leq 2) \approx 0.072 + 0.283 + 0.396 = 0.751
\]

Thus:

\[
P(X > 2) = 1 - 0.751 = 0.249
\]

### Summary of Answers:
A. Probability that none of the people are Independent: \( P(X = 0) \approx 0.072 \)

B. Probability that fewer than 6 are Independent: \( P(X < 6) \approx 0.998 \)

C. Probability that more than 2 people are Independent: \( P(X > 2) \approx 0.249 \)

Would you like more details on any part of the solution?

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Here are 5 questions related to this problem:
1. How does the binomial distribution change if the probability of success increases?
2. What is the cumulative distribution function in binomial probability?
3. How would the results change if the survey size was increased to 50 people?
4. How can the normal approximation be applied to binomial distributions?
5. What is the probability of exactly 6 Independents in the same scenario?

**Tip:** In binomial distributions, probabilities of "at least" or "more than" are often easier to compute using complementary probabilities.

In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 12% of voters are Independent. A survey asked 29 people to identify themselves as Democrat, Republican, or Independent.

A. What is the probability that none of the people are Independent?

B. What is the probability that fewer than 6 are Independent?

C. What is the probability that more than 2 people are Independent?

Explore the binomial probability of voters identifying as Independent in a 29-person survey, with a success rate of 12%. Find out the probabilities for none, fewer than 6, and more than 2 Independents, using key probability formulas and the binomial theorem. Suitable for students in Grades 9-12.

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