https://file.aiquickdraw.com/mathbot/file/1727851205275xo6o25xx.jpg

The image describes a binomial probability problem where the parameters are given as:

- \( n = 10 \) (number of trials),
- \( p = 0.4 \) (probability of success in each trial), 
- \( x = 3 \) (number of successes), 

and the task is to compute \( P(3) \), the probability of exactly 3 successes in 10 independent trials.

### Binomial Probability Formula:
The formula for the binomial probability is given by:

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

where:
- \( \binom{n}{x} \) is the binomial coefficient, which represents the number of ways to choose \( x \) successes from \( n \) trials, and is computed as:

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

### Solution Steps:
For \( n = 10 \), \( p = 0.4 \), and \( x = 3 \):
1. **Compute the binomial coefficient** \( \binom{10}{3} \):
   \[
   \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
   \]
   
2. **Calculate the probability**:
   \[
   P(3) = \binom{10}{3} (0.4)^3 (0.6)^{7}
   \]
   \[
   P(3) = 120 \times (0.4)^3 \times (0.6)^7
   \]
   First, calculate each part:
   - \( (0.4)^3 = 0.064 \)
   - \( (0.6)^7 \approx 0.0279936 \)
   
   Now, multiply everything:
   \[
   P(3) = 120 \times 0.064 \times 0.0279936 \approx 0.2149
   \]

Thus, the probability \( P(3) \approx 0.2149 \), which matches the answer in the image.

Would you like more details on this calculation or any related questions?

### Follow-up questions:
1. How is the binomial distribution used in real-world scenarios?
2. Can you explain why the binomial coefficient is used in this formula?
3. What happens to the binomial distribution as the number of trials increases?
4. How would the result change if \( p \) were closer to 0 or 1?
5. How is the expected value of a binomial distribution calculated?

**Tip**: The sum of all binomial probabilities for a given \( n \) always equals 1 (since one of the possible outcomes must occur).

A binomial probability experiment is conducted with the parameters n = 10, p = 0.4, x = 3. Compute the probability of exactly 3 successes in 10 independent trials.

Learn how to compute the binomial probability of exactly 3 successes in 10 trials with success probability 0.4. Detailed steps include binomial coefficient calculation and applying the binomial formula.

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