percent of adults in a certain country believe that life on other planets is plausible. You randomly select five adults and ask them whether they believe that life on other planets is plausible. The random variable represents the number of adults who believe that life on other planets is plausible. Find the​ mean, variance, and standard deviation of the binomial distribution for the random variable. Interpret the results.
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Part 1
Find the variance of the binomial distribution

To find the variance of a binomial distribution, we need to know the following parameters:

1. **\( n \)**: The number of trials (in this case, the number of adults selected).
2. **\( p \)**: The probability of success (in this context, the probability that an adult believes that life on other planets is plausible).

The formula for the variance of a binomial distribution is:

\[
\text{Variance} = \sigma^2 = n \cdot p \cdot (1 - p)
\]

### Step-by-Step Solution

1. **Given Information:**
   - Number of trials, \( n = 5 \) (since you are selecting 5 adults).
   - Probability of success, \( p \) (probability that an adult believes in life on other planets).

2. **Variance Calculation:**
   - The variance is calculated as:

\[
\sigma^2 = n \cdot p \cdot (1 - p)
\]

To calculate the variance, you need the value of \( p \), the probability of success. If you provide the probability \( p \), I can compute the exact variance.

### If \( p \) is given, for example:

- If \( p = 0.6 \), then:

\[
\sigma^2 = 5 \cdot 0.6 \cdot (1 - 0.6) = 5 \cdot 0.6 \cdot 0.4 = 1.2
\]

Please provide the value of \( p \) to calculate the variance accurately.

Would you like me to proceed with an example, or do you have the specific probability value?

percent of adults in a certain country believe that life on other planets is plausible. You randomly select five adults and ask them whether they believe that life on other planets is plausible. The random variable represents the number of adults who believe that life on other planets is plausible. Find the​ mean, variance, and standard deviation of the binomial distribution for the random variable. Interpret the results.

Explore how to calculate the mean, variance, and standard deviation for a binomial distribution in a real-world scenario where adults are surveyed about life on other planets. Learn key probability concepts and formulas related to binomial distributions, suitable for Grades 10-12 students.

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