According to one​ poll, 42​% of households in a certain region still had landline phone service. Suppose a random sample of 65 households from the region is taken. Complete parts​ (a) through​ (d) below.
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Part 1
a. Find the probability that exactly 28 of the households sampled still have a landline.
The probability that exactly 28 still have a landline is 


This problem can be modeled using the binomial distribution since we have a fixed number of trials (65 households), two possible outcomes (having a landline or not), and a constant probability of success (42%).

The binomial probability formula is:

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

Where:
- \( n = 65 \) (the number of trials, or households)
- \( k = 28 \) (the number of households with landlines we're interested in)
- \( p = 0.42 \) (the probability of a household having a landline)
- \( \binom{n}{k} \) is the binomial coefficient

Now, let's compute the probability using this formula.The probability that exactly 28 of the households sampled still have a landline is approximately 0.098, or 9.8%.

Would you like to dive into more details or explore further? Here are some related questions:

1. How would you calculate the mean and standard deviation for this binomial distribution?
2. What is the probability that fewer than 28 households have a landline?
3. How does the binomial distribution change if the sample size increases?
4. What assumptions must hold true for this to be modeled as a binomial distribution?
5. What is the probability of more than 30 households having a landline?

**Tip:** For large samples, the binomial distribution can often be approximated by a normal distribution.

According to one​ poll, 42​% of households in a certain region still had landline phone service. Suppose a random sample of 65 households from the region is taken. Complete parts​ (a) through​ (d) below. Part 1 a. Find the probability that exactly 28 of the households sampled still have a landline.

Solve a binomial probability problem by calculating the likelihood that exactly 28 out of 65 households have a landline. This problem uses the binomial distribution formula with a success probability of 42%. Suitable for Grades 11-12.

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