A TV executive is interested in the popularity of a particular streaming TV show. She has been told that a whopping 68% of American households would be interested in turing in to a new network version of the show. If this is correct, what is the probability that all 6 of the households in her city being monitored by the TV industry would tune in to the new show? Assume that the 6 households constitute a random sample of American households.

Round your response to at least three decimal places. 

To determine the probability that all 6 of the households being monitored will tune in to the new show, we can again use the binomial probability formula. In this case, we want the probability that all 6 households (i.e., \( k = 6 \)) are interested in the show.

Given:
- The probability of success (a household tuning in) is \( p = 0.68 \).
- The number of trials (households) is \( n = 6 \).
- The number of successes (households tuning in) is \( k = 6 \).

The binomial probability formula is:

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

Since we want all 6 households to tune in, \( k = n = 6 \). This simplifies the formula because \(\binom{n}{n} = 1\), so:

\[
P(X = 6) = 1 \times (0.68)^6 \times (1-0.68)^{6-6}
\]

Calculating:

1. **Calculate \( p^6 \)**:

\[
(0.68)^6 \approx 0.0885
\]

2. **Since \( (1-p)^0 = 1 \)**, the probability simplifies to:

\[
P(X = 6) \approx 0.0885
\]

Therefore, the probability that all 6 households will tune in to the new show is approximately **0.089** (rounded to three decimal places).

---

Do you want more details or have any other questions? Here are some related questions:

1. What is the probability that exactly 5 of the households will tune in to the new show?
2. How does the probability change if the sample size is increased to 10 households?
3. What is the probability that none of the households will tune in to the new show?
4. How would the probability change if the interest in the show increased to 80%?
5. What is the expected number of households that will tune in out of the 6 being monitored?
6. What is the cumulative probability of at least 4 households tuning in to the new show?
7. How would the probability change if the executive monitored only 3 households?
8. Can a normal distribution approximation be used for this problem, and when would it be appropriate?

**Tip:** When calculating probabilities for all successes (or all failures), the binomial coefficient simplifies to 1, making the calculation easier.

Discover how to calculate the probability that all 6 households in a random sample will tune in to a new TV show. Use the binomial probability formula with given probabilities and number of trials. Round the result to three decimal places for accuracy.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation