In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 13% of voters are Independent. A survey asked 34 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 4 are Independent?

Probability = 

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

Probability = 


Given:
- The probability of a voter being Independent, \( p = 0.13 \).
- The number of people surveyed, \( n = 34 \).

This is a binomial probability problem where the probability of exactly \( k \) Independents in \( n \) trials is given by:

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

Let's solve each part.

### Part A: Probability that none of the people are Independent
This is the probability that \( X = 0 \).

\[
P(X = 0) = \binom{34}{0} (0.13)^0 (0.87)^{34} = (0.87)^{34}
\]

Calculating this:

\[
P(X = 0) \approx 0.032
\]

### Part B: Probability that fewer than 4 are Independent
This is the probability that \( X < 4 \), or \( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \).

Each term is calculated as:

1. \( P(X = 0) = (0.87)^{34} \)
2. \( P(X = 1) = \binom{34}{1} (0.13)^1 (0.87)^{33} \)
3. \( P(X = 2) = \binom{34}{2} (0.13)^2 (0.87)^{32} \)
4. \( P(X = 3) = \binom{34}{3} (0.13)^3 (0.87)^{31} \)

Summing these probabilities gives \( P(X < 4) \approx 0.658 \).

### Part C: Probability that more than 2 people are Independent
This is the probability that \( X > 2 \), which can be calculated as:

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

From Part B, we already know \( P(X \leq 2) \approx 0.502 \), so:

\[
P(X > 2) \approx 1 - 0.502 = 0.498
\]

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### Final Answers
A. \( P(X = 0) \approx 0.032 \)

B. \( P(X < 4) \approx 0.658 \)

C. \( P(X > 2) \approx 0.498 \)

Would you like a more detailed breakdown of the calculations? 

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**Questions for Practice:**
1. How would the probabilities change if the survey size were 50 people instead of 34?
2. What if the percentage of Independents was 20% instead of 13%?
3. How is the binomial distribution related to other probability distributions?
4. How would we calculate the expected number of Independents in a survey of 34 people?
5. What is the standard deviation of the number of Independents in this survey?

**Tip:** In a binomial distribution, for probabilities of events with many trials, use a normal approximation when \( np \) and \( n(1 - p) \) are both greater than 5.

In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 13% of voters are Independent. A survey asked 34 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 4 are Independent?

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

Calculate probabilities for Independent voters in a 34-person survey using binomial distribution. Determine the likelihood of having zero, fewer than 4, or more than 2 Independent voters, assuming a 13% probability for Independents. Perfect for high school students studying probability and binomial distributions.

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