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.