Using the binomial probability distribution, when p=0.30 and n=14 what is P(X=4)? 

To find the probability \( P(X = 4) \) using the binomial probability distribution formula, we will use the following formula:

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

Where:
- \( n = 14 \) (number of trials),
- \( p = 0.30 \) (probability of success on a single trial),
- \( X = k = 4 \) (number of successes).

We need to calculate \( \binom{n}{k} \), which is the number of combinations (the binomial coefficient), followed by the rest of the formula:

\[
P(X = 4) = \binom{14}{4} (0.30)^4 (1 - 0.30)^{14-4}
\]

\[
P(X = 4) = \binom{14}{4} (0.30)^4 (0.70)^{10}
\]

Now, let's compute this.

First, calculate \( \binom{14}{4} \):

\[
\binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001
\]

Next, calculate the terms:

\[
0.30^4 = 0.0081
\]
\[
0.70^{10} = 0.0282475
\]

Now, calculate \( P(X = 4) \):

\[
P(X = 4) = 1001 \times 0.0081 \times 0.0282475
\]

Let me compute this final value.The probability \( P(X = 4) \) when \( p = 0.30 \) and \( n = 14 \) is approximately \( 0.229 \), or 22.9%.

Would you like more details, or do you have any other questions?

Here are some related questions to expand on this topic:
1. How would the probability change if \( p \) increased to 0.40?
2. What is the cumulative probability \( P(X \leq 4) \)?
3. How can we interpret binomial probabilities in real-world scenarios?
4. What is the expected value and variance for this binomial distribution?
5. How do you calculate \( P(X \geq 5) \) using complementary probabilities?

**Tip:** The sum of all probabilities in a binomial distribution for a given \( n \) must always equal 1.

Using the binomial probability distribution, when p=0.30 and n=14 what is P(X=4)?

Explore how to calculate the binomial probability P(X = 4) when p=0.30 and n=14. This detailed step-by-step solution explains the binomial distribution formula, including combinatorics and probability calculations. Suitable for high school students in Grades 10-12.

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