SOLUTION: In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 10% of voters are Independent. A survey asked 23 people to identify

Question 1198689: In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 10% of voters are Independent. A survey asked 23 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 = ?
Answer by textot(100) (Show Source): You can put this solution on YOUR website!
Certainly, let's calculate the probabilities.
**Assumptions:**
* The number of Independents follows a binomial distribution.
* The probability of a voter being Independent is constant (p = 0.10).
* The voters are independent of each other.
**A. Probability that none of the people are Independent**
* In this case, the number of successes (Independent voters) is 0.
* Using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
where:
* n = sample size = 23
* k = number of successes = 0
* p = probability of success (Independent) = 0.10
* q = probability of failure (Not Independent) = 1 - p = 0.90
* P(X = 0) = (23C0) * (0.10)^0 * (0.90)^(23-0)
= 1 * 1 * (0.90)^23
≈ 0.0948
**Therefore, the probability that none of the people are Independent is approximately 0.0948.**
**B. Probability that fewer than 6 are Independent**
* We need to find the probability of 0, 1, 2, 3, 4, or 5 Independents.
* P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
* Calculate each probability using the binomial probability formula and sum them up.
* Using a calculator or statistical software, we can find:
P(X < 6) ≈ 0.9917
**Therefore, the probability that fewer than 6 people are Independent is approximately 0.9917.**
**C. Probability that more than 2 people are Independent**
* We need to find the probability of 3, 4, 5, ..., 23 Independents.
* P(X > 2) = 1 - P(X ≤ 2)
= 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
* Calculate each probability using the binomial probability formula and sum them up.
* Using a calculator or statistical software, we can find:
P(X > 2) ≈ 0.0593
**Therefore, the probability that more than 2 people are Independent is approximately 0.0593.**
I hope this helps! Let me know if you have any other questions.

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