In a binomial probability experiment, the probability of exactly $x$ successes in $n$ independent trials, with each trial having a success probability $p$, is given by the binomial probability formula:

$P(x) = \binom{n}{x} p^x (1 - p)^{n - x}$

Where:

• $n$ is the number of trials,
• $x$ is the number of successes,
• $p$ is the probability of success on each trial,
• $\binom{n}{x}$ is the binomial coefficient, which is calculated as:

$\binom{n}{x} = \frac{n!}{x!(n - x)!}$

Now, plug in the given values:

• $n = 30$,
• $p = 0.04$,
• $x = 2$.

We first calculate the binomial coefficient:

$\binom{30}{2} = \frac{30!}{2!(30 - 2)!} = \frac{30 \times 29}{2 \times 1} = 435$

Now, compute the probability:

$P(2) = 435 \times (0.04)^2 \times (1 - 0.04)^{30 - 2}$ $P(2) = 435 \times 0.0016 \times (0.96)^{28}$ $P(2) = 435 \times 0.0016 \times 0.3725$ $P(2) \approx 435 \times 0.000596 = 0.25926$

Finally, rounding to four decimal places:

$P(2) \approx 0.2593$

Thus, the probability of 2 successes is approximately 0.2593.

Would you like a breakdown of the steps or have any further questions?

Here are 5 questions to expand your understanding:

1. How does the binomial distribution change with different probabilities $p$?
2. What is the expected number of successes in this scenario?
3. How would the probability change if we wanted 3 successes instead of 2?
4. How do you calculate cumulative probabilities for $P(x \leq 2)$?
5. What happens to the binomial distribution as $n$ increases while keeping $p$ constant?

Tip: When working with binomial probabilities, it's helpful to familiarize yourself with using a scientific calculator or software for large factorials.