A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.
n equals 12​, p equals 0.4​, x less than or equals 4
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Part 1
The probability of x less than or equals 4 successes is 
  
enter your response here. ​(Round to four decimal places as​ needed.)

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 \( \sigma^2 = np(1 - p) \).

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n equals 12, p equals 0.4, x less than or equals 4. The probability of x less than or equals 4 successes is ______ (Round to four decimal places as needed).

Compute the probability of getting up to 4 successes in 12 independent trials with a success probability of 0.4. Learn how to apply the binomial probability formula and understand the cumulative distribution for x ≤ 4. Suitable for Grades 10-12 students.

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