The binomial probability formula is given by:

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

Where:

• $n$ = 12 (number of trials)
• $p$ = 0.4 (probability of success)
• $x \leq 4$ (we want the cumulative probability for $x = 0, 1, 2, 3, 4$)

The cumulative probability of $x \leq 4$ is the sum of the probabilities for $x = 0, 1, 2, 3, 4$:

$P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$

Let’s compute these probabilities step by step. I'll calculate this now.The probability of getting $x \leq 4$ successes in 12 independent trials, with a probability of success $p = 0.4$, is approximately 0.4382 (rounded to four decimal places).

Do you have any questions or would you like further details on this?

Here are five related questions you might find useful:

1. How does changing $p$ affect the binomial distribution?
2. What happens to the probability when $n$ increases?
3. How do we calculate the mean and variance of a binomial distribution?
4. What is the probability of exactly 4 successes in this experiment?
5. How does the binomial distribution differ from the normal distribution?

Tip: The mean of a binomial distribution is $\mu = np$, and the variance is $\sigma^2 = np(1 - p)$.