In the most recent election, 19 percent of all eligible college students voted. If a random sample of 20 students were surveyed, find the probability that exactly half voted in the election.




This is a **binomial probability** problem, where we need to find the probability that exactly half of the students (10 out of 20) voted in the election, given that the probability of any student voting is 19%, or \( p = 0.19 \).

We can use the **binomial probability formula**:

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

Where:
- \( n = 20 \) (number of trials, i.e., students),
- \( k = 10 \) (desired number of successes, i.e., students who voted),
- \( p = 0.19 \) (probability of voting),
- \( 1 - p = 0.81 \) (probability of not voting).

### Step-by-Step Calculation:

1. **Calculate the binomial coefficient** \( \binom{20}{10} \), which gives the number of ways to choose 10 students out of 20.
2. **Raise \( p \) to the power of \( k \)** (i.e., \( 0.19^{10} \)).
3. **Raise \( 1 - p \) to the power of \( n - k \)** (i.e., \( 0.81^{10} \)).
4. Multiply these together to get the probability.

Let’s calculate that now.The probability that exactly half (10 out of 20) of the students voted in the election is approximately \( 0.00138 \), or 0.138%.

Would you like more details on how this was calculated, or do you have any further questions?

Here are 5 related questions to explore:

1. What is the probability that none of the 20 students voted?
2. What is the probability that at least one student voted?
3. What is the probability that fewer than 5 students voted?
4. How does the probability change if the voting rate increases to 30%?
5. How many students would need to be surveyed for a 50% chance of exactly half voting?

**Tip:** When dealing with binomial distributions, it's useful to understand the mean and standard deviation to predict the typical number of successes.

In the most recent election, 19 percent of all eligible college students voted. If a random sample of 20 students were surveyed, find the probability that exactly half voted in the election.

This problem explores how to calculate the probability that exactly half of 20 students voted in an election, given a 19% voting rate, using the binomial probability formula. A step-by-step solution with key mathematical concepts such as binomial distribution and probability. Suitable for advanced high school or introductory college students.

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