A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.
nequals12​, pequals0.4​, xequals5
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Upper P left parenthesis 5 right parenthesisequals
  
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​(Do not round until the final answer. Then round to four decimal places as​ needed.)

In a binomial probability experiment, the probability of exactly \( x \) successes in \( n \) independent trials 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 in a single trial
- \( \binom{n}{x} \) is the binomial coefficient, which is calculated as:

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

### Given:
- \( n = 12 \)
- \( p = 0.4 \)
- \( x = 5 \)

First, calculate the binomial coefficient:

\[
\binom{12}{5} = \frac{12!}{5!(12 - 5)!} = \frac{12!}{5!7!}
\]

Now, calculate the binomial probability:

\[
P(5) = \binom{12}{5} (0.4)^5 (1 - 0.4)^{12 - 5} = \binom{12}{5} (0.4)^5 (0.6)^7
\]

Let me calculate this.The probability of obtaining exactly 5 successes in 12 independent trials with a success probability of 0.4 is approximately \( P(5) = 0.2270 \).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the probability change if \( p \) increased to 0.6?
2. What is the probability of getting 6 successes instead of 5?
3. How is the binomial distribution related to the normal distribution for large \( n \)?
4. Can you find the cumulative probability of getting 5 or fewer successes?
5. How would the binomial probability change if the number of trials, \( n \), was 15 instead of 12?

**Tip:** In binomial distributions, the mean is given by \( n \cdot p \) and the variance by \( n \cdot p \cdot (1 - p) \), useful for quick insights into the distribution's behavior.

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 = 12, p = 0.4, x = 5

Calculate the probability of obtaining 5 successes in 12 independent trials with a success rate of 0.4 using the binomial distribution formula. Learn about key probability concepts, including binomial coefficients and success probabilities. Suitable for high school students.

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