In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 11% of voters are Independent. A survey asked 40 people to identify themselves as Democrat, Republican, or Independent.

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

Probability = 0.0037

B. What is the probability that fewer than 5 are Independent?

Probability = 0.923

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

Probability = 


Let's break down the problem with a binomial distribution approach:

1. **Given**:
   - Probability of a person being an Independent voter, \( p = 0.11 \)
   - Number of people surveyed, \( n = 40 \)

   For each part of the question, we will calculate the probability using the binomial distribution formula:
   \[
   P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
   \]
   where \( X \) represents the number of Independent voters in the sample, and \( k \) is the specific count for which we want the probability.

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### Part C: Probability that more than 2 people are Independent

To find \( P(X > 2) \), we can use:
\[
P(X > 2) = 1 - P(X \leq 2)
\]
where \( P(X \leq 2) \) represents the probability of having 0, 1, or 2 Independent voters.

1. **Calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \)** using the binomial formula.

2. **Sum these probabilities**:
   \[
   P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
   \]

3. **Subtract from 1**:
   \[
   P(X > 2) = 1 - P(X \leq 2)
   \]

Let's go through these calculations:

1. **\( P(X = 0) \):**
   \[
   P(X = 0) = \binom{40}{0} (0.11)^0 (0.89)^{40} \approx 0.0037
   \]

2. **\( P(X = 1) \):**
   \[
   P(X = 1) = \binom{40}{1} (0.11)^1 (0.89)^{39} \approx 0.0184
   \]

3. **\( P(X = 2) \):**
   \[
   P(X = 2) = \binom{40}{2} (0.11)^2 (0.89)^{38} \approx 0.0547
   \]

4. **Sum of probabilities** \( P(X \leq 2) \):
   \[
   P(X \leq 2) = 0.0037 + 0.0184 + 0.0547 = 0.0768
   \]

5. **Final answer for \( P(X > 2) \):**
   \[
   P(X > 2) = 1 - 0.0768 = 0.9232
   \]

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### Conclusion
The probability that more than 2 people are Independent is approximately **0.9232**.

Would you like more details on any part of this solution, or do you have other questions? 

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Here are 5 related questions for further exploration:
1. What would the probability be if only 20 people were surveyed?
2. How would the probability change if the percentage of Independents were 15%?
3. Can the binomial distribution be used if the survey had a larger sample size, say 100?
4. How would we interpret this probability in terms of confidence for predicting voter trends?
5. What is the cumulative probability distribution of this survey?

**Tip:** Remember that when using the binomial distribution, the assumptions are that each trial is independent and has only two possible outcomes (success or failure). This is crucial for accurate calculations.

In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 11% of voters are Independent. A survey asked 40 people to identify themselves as Democrat, Republican, or Independent.

A. What is the probability that none of the people are Independent?
Probability = 0.0037

B. What is the probability that fewer than 5 are Independent?
Probability = 0.923

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

Explore the probabilities of independent voters in a survey of 40 people using binomial distribution. This problem calculates the probabilities that none, fewer than 5, and more than 2 respondents identify as Independent. Key formulas include binomial probability and complement rule, making it suitable for students in Grades 10-12.

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