The problem involves a binomial probability, where we are asked to find \( P(x = 3) \), which represents the probability of getting exactly 3 successes in \( n = 6 \) independent trials, each with a success probability \( p = 0.8 \).

The binomial probability formula is:

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

Where:
- \( n = 6 \) (the number of trials),
- \( k = 3 \) (the number of successes),
- \( p = 0.8 \) (the probability of success in each trial),
- \( 1 - p = 0.2 \) (the probability of failure).

Let's calculate \( P(x = 3) \).

1. Calculate \( \binom{n}{k} \), which is the number of combinations of 6 trials taken 3 at a time:

\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

2. Now, use the binomial probability formula:

\[
P(x = 3) = 20 \cdot (0.8)^3 \cdot (0.2)^3
\]

I'll compute this now.The probability of getting exactly 3 successes in 6 trials, with a success probability of \( p = 0.8 \), is approximately \( P(x = 3) = 0.08192 \).

Would you like further details or have any questions?

Here are five related questions:
1. How do we calculate binomial probabilities for different values of \( x \)?
2. What is the cumulative probability for \( P(x \leq 3) \)?
3. How does the binomial distribution change with different values of \( n \) and \( p \)?
4. What is the expected value and variance of a binomial distribution?
5. How would you calculate the probability for more than 3 successes?

**Tip**: The binomial probability formula can be used to model any situation with a fixed number of independent trials and two possible outcomes, such as success/failure scenarios.

Learn how to calculate the binomial probability P(x = 3) for n = 6 and p = 0.8 using the binomial probability formula. This detailed solution covers key probability concepts and combinatorial methods, ideal for students in Grades 9-12.

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