Based on a​ survey, 32% of likely voters would be willing to vote by internet instead of the​ in-person traditional method of voting. For each of the​ following, assume that 15 likely voters are randomly selected. Complete parts​ (a) and​ (b) below.
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Part 1
a. What is the probability that exactly 12 of those selected would do internet​ voting?

To solve this, we will use the **binomial probability formula**. This formula is used to determine the probability of obtaining exactly \( k \) successes in \( n \) independent Bernoulli trials, where each trial has two outcomes: success (with probability \( p \)) and failure (with probability \( 1-p \)).

The binomial probability formula is:

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

where:
- \( n \) is the number of trials (in this case, 15 likely voters),
- \( k \) is the number of successes (in this case, 12 voters who choose internet voting),
- \( p \) is the probability of success on a single trial (in this case, 32%, or 0.32),
- \( \binom{n}{k} \) is the number of ways to choose \( k \) successes from \( n \) trials, calculated as \( \frac{n!}{k!(n-k)!} \).

### Given:
- \( n = 15 \),
- \( k = 12 \),
- \( p = 0.32 \).

Let me calculate the exact probability.The probability that exactly 12 out of 15 selected likely voters would choose internet voting is approximately **0.000165** (or 0.0165%).

Would you like further details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the probability that fewer than 12 voters would choose internet voting?
2. How would the probability change if 20 voters were randomly selected instead of 15?
3. What is the probability that at least 10 voters would choose internet voting?
4. How does changing the probability of success from 32% to 50% affect the result?
5. Can you calculate the expected number of voters who would prefer internet voting?

**Tip:** Binomial distribution can be used to model scenarios with fixed trials and independent outcomes, such as flipping a coin or conducting surveys.

Based on a survey, 32% of likely voters would be willing to vote by internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected. Complete parts (a) and (b) below. Part 1: a. What is the probability that exactly 12 of those selected would do internet voting?

Explore the probability of exactly 12 out of 15 likely voters choosing internet voting. This problem uses binomial probability with a success rate of 32%, helping to illustrate key probability concepts. Suitable for Grades 9-12.

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