Assume that when adults with smartphones are randomly​ selected, 55​% use them in meetings or classes. If 25 adult smartphone users are randomly​ selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.
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Part 1
The probability is 
  
enter your response here.

We can solve this problem using the **binomial probability formula** since we are dealing with a fixed number of trials (25 adults), two possible outcomes (using a smartphone in meetings or not), and a constant probability (55%).

The binomial probability formula is:

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

where:
- \( P(X = k) \) is the probability of exactly \( k \) successes,
- \( n \) is the number of trials (25 adults),
- \( k \) is the number of successes (15 adults using smartphones),
- \( p \) is the probability of success on a single trial (0.55),
- \( 1 - p \) is the probability of failure (0.45),
- \( \binom{n}{k} \) is the binomial coefficient: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Now let's calculate:

\[
P(X = 15) = \binom{25}{15} (0.55)^{15} (0.45)^{10}
\]

First, compute \( \binom{25}{15} \):

\[
\binom{25}{15} = \frac{25!}{15!(25-15)!} = \frac{25!}{15!10!} = 3268760
\]

Now plug this into the formula:

\[
P(X = 15) = 3268760 \times (0.55)^{15} \times (0.45)^{10}
\]

Using a calculator:

\[
P(X = 15) \approx 3268760 \times 0.00258 \times 0.00253
\]
\[
P(X = 15) \approx 21.36\%
\]

Thus, the probability that exactly 15 out of 25 adults use their smartphones in meetings or classes is approximately **0.2136** or **21.36%**.

Do you want further details or have any questions?

### Related questions:
1. What is the probability that fewer than 10 adults use their smartphones in meetings or classes?
2. How would the probability change if only 10 adults were selected instead of 25?
3. What is the expected number of adults using smartphones in meetings from a group of 25?
4. How would you calculate the probability that more than 20 adults use their smartphones in meetings?
5. What is the variance and standard deviation for the number of adults using smartphones?

**Tip:** The sum of binomial probabilities over all possible outcomes equals 1, as they represent all possible results of the experiment.

Assume that when adults with smartphones are randomly selected, 55% use them in meetings or classes. If 25 adult smartphone users are randomly selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.

This problem demonstrates how to calculate the probability that exactly 15 out of 25 adults use smartphones in meetings or classes using the binomial distribution. Learn to apply the binomial probability formula step-by-step, including the binomial coefficient calculation. Suitable for students in Grades 10-12.

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