​(c) In a random sample of 15 women 18 years old or​ older, find the probability that at least 5 believe that the minimum driving age should be 18.
The probability is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

To solve this problem, we need to use the **binomial probability distribution**, because we are dealing with a situation where there are two outcomes: either a woman believes the minimum driving age should be 18 (success) or she doesn't (failure). The number of women in the sample is fixed, and we're finding the probability that at least 5 women believe the minimum driving age should be 18.

Let's break it down into steps:

### 1. Define the variables
- \( n = 15 \): The number of trials (women in the sample).
- \( k \geq 5 \): We're looking for the probability of getting at least 5 successes.
- \( p \): The probability of success on a single trial (i.e., the probability that a woman believes the minimum driving age should be 18).

Since you haven't provided \( p \) (the probability that a woman believes the minimum driving age should be 18), let's assume \( p \) is a known value (or you can clarify it). For now, I'll proceed without a specific value for \( p \), but the steps remain the same:

### 2. Probability of "at least 5" successes
The probability of getting at least 5 successes can be written as:
\[
P(X \geq 5) = 1 - P(X < 5)
\]
where \( X \) is the number of women who believe the minimum driving age should be 18.

We can calculate this by finding the sum of the probabilities for 0, 1, 2, 3, and 4 successes using the binomial formula:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
for \( k = 0, 1, 2, 3, 4 \), and then subtract the sum of these probabilities from 1.

### 3. Compute and round to four decimal places
Once we have the probability \( p \), we can plug it into the binomial probability formula and compute the sum for the cases \( k = 0, 1, 2, 3, 4 \), then subtract from 1.

If you can provide the probability \( p \) or any additional information, I can give you the exact answer!

Do you want more details on the calculation or assumptions? 

### Expanding questions:
1. What is the value of \( p \) in this scenario?
2. How do we use binomial probability distribution in real-life situations?
3. What other distributions could be used in similar problems, and why?
4. How does changing \( p \) affect the outcome of the probability?
5. Can we use a normal approximation for this binomial distribution?

**Tip:** For large sample sizes, the binomial distribution can often be approximated by a normal distribution, especially when \( np \) and \( n(1 - p) \) are both greater than 5.

In a random sample of 15 women 18 years old or older, find the probability that at least 5 believe that the minimum driving age should be 18. (Round to four decimal places as needed.)

Learn how to calculate the binomial probability that at least 5 women out of a sample of 15 believe the minimum driving age should be 18. This problem involves using the binomial probability formula and understanding key concepts in probability theory. Detailed step-by-step explanation provided. Suitable for advanced high school or college students.

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