Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places.

P(X=6)
, n=9
, p=0.5

Given that the random variable \( X \) follows a binomial distribution, we can calculate the probability \( P(X = 6) \) using the binomial probability formula:

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

Where:
- \( n = 9 \) is the number of trials,
- \( k = 6 \) is the number of successes,
- \( p = 0.5 \) is the probability of success.

### Step-by-step calculation:
1. **Compute the binomial coefficient \( \binom{n}{k} \):**
\[
\binom{9}{6} = \frac{9!}{6!(9 - 6)!} = \frac{9!}{6!3!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
\]

2. **Calculate the probability:**
\[
P(X = 6) = 84 \times (0.5)^6 \times (0.5)^3
\]
\[
P(X = 6) = 84 \times (0.5)^9 = 84 \times 0.001953125 = 0.1641
\]

Thus, the probability \( P(X = 6) \) is approximately **0.1641**.

### Do you want more details or have any questions?

Here are some relative questions:
1. What is the probability of getting exactly 5 successes?
2. How does the binomial distribution change with different values of \( p \)?
3. How do you compute the cumulative probability \( P(X \leq k) \)?
4. What is the expected value and variance for this binomial distribution?
5. How would the probability change if \( p \) was 0.6 instead of 0.5?

**Tip**: The binomial distribution is symmetric when \( p = 0.5 \), making it easier to calculate probabilities for certain values.

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X=6), n=9, p=0.5

Learn how to calculate the binomial probability P(X=6) given a binomial distribution with 9 trials and a probability of success of 0.5. The detailed step-by-step solution explains how to use the binomial probability formula, including calculating the binomial coefficient. Suitable for Grades 10-12.

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