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This problem is based on binomial probability, which applies when there are a fixed number of independent trials (here, the number of 18- to 29-year-olds surveyed), each with two possible outcomes (owning or not owning a tablet), and the probability of success is the same for each trial.

### Problem Summary:
- 36% of 18- to 29-year-olds own a tablet (this is the probability of success, \( p = 0.36 \)).
- We are surveying 6 people (so \( n = 6 \)).
- For each sub-question, we are calculating the probability based on this binomial distribution.

The probability mass function (PMF) for a binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}
\]
Where:
- \( P(X = k) \) is the probability of exactly \( k \) successes.
- \( n \) is the number of trials (in this case, 6).
- \( p \) is the probability of success (36% or 0.36).
- \( k \) is the number of successes we want to find the probability for.
- \( \binom{n}{k} \) is the binomial coefficient, which calculates the number of ways to choose \( k \) successes from \( n \) trials.

### Part (a):
This has already been completed, and the probability of exactly 3 people owning a tablet out of 6 is provided as 0.1033.

### Part (b):
For this part, we are asked to find the probability that **all six** people own a tablet, i.e., \( P(X = 6) \).

Substitute the values into the binomial formula:
\[
P(X = 6) = \binom{6}{6} \cdot (0.36)^6 \cdot (1 - 0.36)^0
\]
Since \( \binom{6}{6} = 1 \) and \( (1 - 0.36)^0 = 1 \), the formula simplifies to:
\[
P(X = 6) = (0.36)^6
\]

Let me calculate this probability for you.The probability that all six people surveyed will own a tablet is approximately \( 0.0022 \) or 0.22%.

Would you like more details or have any further questions? Here are 5 related questions for deeper exploration:

1. How would the probability change if 10 people were surveyed instead of 6?
2. What is the probability that exactly 5 out of the 6 people own a tablet?
3. How does increasing the probability of tablet ownership to 50% affect the results?
4. What is the cumulative probability that at least 4 people in the survey own a tablet?
5. Can you explain how the binomial coefficient \( \binom{n}{k} \) affects the probability calculations?

**Tip:** In binomial distributions, as the number of trials increases, the distribution becomes more symmetric, especially when \( p \) is close to 0.5.

A recent survey reported that 36% of 18- to 29-year-olds in a certain country own tablets. Using the binomial distribution, complete parts (a) through (e) below.
a. What is the probability that in the next six 18- to 29-year-olds surveyed, four will own a tablet?
b. What is the probability that in the next six 18- to 29-year-olds surveyed, all six will own a tablet?

Calculate the binomial probability that 36% of surveyed 18- to 29-year-olds own a tablet. Explore the probability that out of 6 people surveyed, 4 or all 6 will own a tablet. Suitable for students in Grades 11-12 learning probability concepts.

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