according to  a recent survey, 80% of all teenagers own a smartphone. if we select 10 teenagers at random, find the probabilit ythat exactly 6 teenagers own a smartphone. 



Express your answer to three decimal placse, i.e 0.XXX

This problem involves finding the probability of exactly 6 teenagers owning a smartphone out of 10, given that 80% (or 0.8) of all teenagers own a smartphone. This is a binomial probability problem because we have a fixed number of trials (10 teenagers), two possible outcomes (own a smartphone or not), and a constant probability of success (0.8).

The binomial probability formula is:

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

Where:
- \( n = 10 \) (number of trials or teenagers),
- \( k = 6 \) (number of successes, i.e., teenagers who own a smartphone),
- \( p = 0.8 \) (probability of owning a smartphone),
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials.

Now, we can calculate the probability:

\[
P(X = 6) = \binom{10}{6} (0.8)^6 (0.2)^4
\]

First, compute \( \binom{10}{6} \):

\[
\binom{10}{6} = \frac{10!}{6!(10 - 6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

Now compute the powers of 0.8 and 0.2:

\[
(0.8)^6 = 0.262144 \quad \text{and} \quad (0.2)^4 = 0.0016
\]

Now, put everything together:

\[
P(X = 6) = 210 \times 0.262144 \times 0.0016
\]

Let me calculate that.The probability that exactly 6 out of 10 randomly selected teenagers own a smartphone is approximately \( 0.088 \) (rounded to three decimal places).

Do you want more details on this calculation or have any other questions?

Here are 5 related questions:
1. How would the probability change if the number of teenagers was 15?
2. What is the probability that fewer than 5 teenagers own a smartphone?
3. What is the probability that more than 8 teenagers own a smartphone?
4. How does the probability change if 90% of teenagers own a smartphone instead of 80%?
5. What is the expected number of teenagers who own a smartphone out of 10?

**Tip:** In binomial probability problems, it's important to clearly identify the probability of success (p) and the number of trials.

According to a recent survey, 80% of all teenagers own a smartphone. If we select 10 teenagers at random, find the probability that exactly 6 teenagers own a smartphone. Express your answer to three decimal places, i.e. 0.XXX.

This detailed solution explains how to find the probability that exactly 6 out of 10 randomly selected teenagers own a smartphone, given that 80% of teenagers own one. It uses binomial distribution concepts and formulas, suitable for students in Grades 10-12.

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