A random variable X has a binomial distribution with p=0.3 and n=10. Determine the probability P(X=7) correct to 3 decimal places.

To determine the probability \( P(X = 7) \) for a binomial random variable \( X \) with parameters \( p = 0.3 \) and \( n = 10 \), we can use the binomial probability formula:

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

where:
- \( n = 10 \) is the number of trials,
- \( k = 7 \) is the number of successes,
- \( p = 0.3 \) is the probability of success on each trial.

Let's calculate it step by step.

1. **Calculate the binomial coefficient** \( \binom{10}{7} \):

\[
\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7! \cdot 3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

2. **Calculate the probability**:

\[
P(X = 7) = 120 \times (0.3)^7 \times (0.7)^{3}
\]

Now, let's compute this.

\[
P(X = 7) = 120 \times (0.0002187) \times (0.343)
\]

\[
P(X = 7) \approx 120 \times 0.0000749501 = 0.00899401
\]

Rounding to three decimal places:

\[
P(X = 7) \approx 0.009
\]

Thus, the probability \( P(X = 7) \) is approximately **0.009**.

Would you like more details or have any other questions? Here are some related questions:

1. How does changing \( p \) affect the binomial distribution?
2. What is the probability \( P(X \leq 7) \) for the given parameters?
3. Can you explain the cumulative distribution function for a binomial variable?
4. How would the probability change if \( n \) were increased to 15?
5. How can the binomial distribution be approximated by a normal distribution?

**Tip:** For small \( p \) and large \( n \), the binomial distribution can be approximated by a Poisson distribution with \( \lambda = np \).

Learn how to calculate the probability P(X = 7) for a binomial random variable with parameters p = 0.3 and n = 10. Step-by-step solution using binomial probability formula. Suitable for advanced high school students studying probability theory.

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