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The problem asks for the probability that a binomial experiment with probability of success \( p = 0.21 \) and \( n = 9 \) trials results in 2 or fewer successes.

### Solution Approach:
We need to calculate \( P(X \leq 2) \), where \( X \) is the binomial random variable with parameters \( n = 9 \) and \( p = 0.21 \).

The formula for the probability of exactly \( k \) successes in a binomial distribution is given by:

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

For this problem, we will calculate the cumulative probability for \( k = 0, 1, 2 \).

1. **For \( k = 0 \):**
\[
P(X = 0) = \binom{9}{0} (0.21)^0 (1 - 0.21)^9 = 1 \times (0.79)^9 = 0.1331
\]

2. **For \( k = 1 \):**
\[
P(X = 1) = \binom{9}{1} (0.21)^1 (1 - 0.21)^8 = 9 \times 0.21 \times (0.79)^8 = 9 \times 0.21 \times 0.2258 = 0.4272
\]

3. **For \( k = 2 \):**
\[
P(X = 2) = \binom{9}{2} (0.21)^2 (1 - 0.21)^7 = 36 \times (0.21)^2 \times (0.79)^7 = 36 \times 0.0441 \times 0.2853 = 0.4534
\]

### Total Probability:
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]
\[
P(X \leq 2) = 0.1331 + 0.4272 + 0.4534 = 1.0137
\]

After rounding to three decimal places:
\[
P(X \leq 2) \approx 1.014
\]

Would you like further details on any step? Here are some additional questions that could expand your understanding:

1. How do we interpret the binomial coefficient in the binomial probability formula?
2. What is the difference between binomial distribution and other probability distributions like Poisson or normal distribution?
3. How does changing the number of trials \(n\) affect the binomial probability distribution?
4. How is the complement rule used in calculating binomial probabilities?
5. How do we calculate cumulative binomial probabilities for any \( k \)-value larger than 2?

**Tip**: When working with binomial distributions, using a cumulative probability table or a binomial probability calculator can save time for large values of \( n \).

A binomial experiment with probability of success p=0.21 and n=9 trials is conducted. What is the probability that the experiment results in 2 or fewer successes?

Calculate the probability of obtaining 2 or fewer successes in a binomial experiment with p=0.21 and n=9 trials. This detailed solution covers binomial distribution, cumulative probability, and step-by-step calculations.

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