We can solve this problem using the binomial probability formula:

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

Where:

• $n = 12$ is the number of trials,
• $k = 4$ is the number of successes,
• $p = 0.54$ is the probability of success on a single trial.

Step 1: Calculate the binomial coefficient

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$ Substitute $n = 12$ and $k = 4$:

$\binom{12}{4} = \frac{12!}{4!(12 - 4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$

Step 2: Calculate the binomial probability

Substitute the values into the formula: $P(X = 4) = 495 \times (0.54)^4 \times (1 - 0.54)^{12 - 4}$ $P(X = 4) = 495 \times (0.54)^4 \times (0.46)^8$

Now, let's compute the probability.The probability of getting exactly 4 successes out of 12 trials, with a success probability of $p = 0.54$, is approximately 0.0844.

Would you like further clarification or details on this? Here are some related questions:

1. What is the probability of getting more than 4 successes?
2. How would the probability change if the success probability was 0.6?
3. What is the expected number of successes in this binomial distribution?
4. How do you calculate cumulative probabilities for a binomial distribution?
5. How does the binomial distribution differ from the normal distribution?

Tip: When working with binomial distributions, the expected value is given by $E(X) = n \times p$, which can help estimate the center of the distribution.