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This problem is asking us to calculate the probability of exactly \( k = 1 \) successes in a binomial distribution with \( n = 13 \) trials and a probability of success \( p = 0.22 \) on each trial. We can use the binomial probability formula:

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

Where:
- \( n = 13 \) (number of trials),
- \( k = 1 \) (number of successes),
- \( p = 0.22 \) (probability of success on a single trial),
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials, and is computed as \( \frac{n!}{k!(n-k)!} \).

Let's calculate the value step by step:

### 1. Binomial coefficient:
\[
\binom{13}{1} = \frac{13!}{1!(13-1)!} = 13
\]

### 2. Compute \( p^k \) and \( (1 - p)^{n-k} \):
\[
p^1 = 0.22
\]
\[
(1 - p)^{12} = (0.78)^{12}
\]

### 3. Putting it all together:
\[
P(X = 1) = 13 \times 0.22 \times (0.78)^{12}
\]

I will now calculate this probability.

The probability of getting exactly \( k = 1 \) success in 13 trials, with a probability of success on a single trial of \( p = 0.22 \), is approximately:

\[
P(X = 1) = 0.1450
\]

Would you like any further details or have any questions about the solution?

Here are 5 related questions to explore:

1. How does the probability change if the number of trials \( n \) increases?
2. What is the probability of getting exactly 0 successes for the same scenario?
3. How can we calculate the cumulative probability of getting at most 2 successes?
4. What is the mean and variance of this binomial distribution?
5. How would the result change if the probability of success \( p \) were higher?

**Tip:** The binomial coefficient \(\binom{n}{k}\) represents the number of different ways \( k \) successes can occur out of \( n \) trials, and it scales the probability calculation based on these combinations.

Assume that a procedure yields a binomial distribution with a trial repeated n = 13 times. Use either the binomial probability formula (or technology) to find the probability of k = 1 successes given the probability p = 0.22 of success on a single trial. Report answer accurate to 4 decimal places. P(X = k) = 

Learn how to calculate the probability of exactly 1 success in 13 trials of a binomial distribution with a success probability of 0.22 per trial. The solution includes key formulas and step-by-step calculations. Suitable for high school students in Grades 9-12.

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