Assume that a procedure yields a binomial distribution with a trial repeated n=9 times. Use either the binomial probability formula (or technology) to find the probability of k=0 successes given the probability p=.21 of success on a single trial.

To find the probability of \( k = 0 \) successes in a binomial distribution, we use the **binomial probability formula**:

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

Where:
- \( n = 9 \) (number of trials),
- \( k = 0 \) (number of successes),
- \( p = 0.21 \) (probability of success on a single trial),
- \( \binom{n}{k} \) is the binomial coefficient.

For \( k = 0 \), the formula simplifies as follows:

\[
P(0) = \binom{9}{0} (0.21)^0 (1 - 0.21)^9
\]

Calculating the components:

- \( \binom{9}{0} = 1 \)
- \( (0.21)^0 = 1 \)
- \( (1 - 0.21) = 0.79 \)

Now compute \( (0.79)^9 \):

\[
P(0) = 1 \times 1 \times (0.79)^9
\]

Let me calculate \( (0.79)^9 \).The probability of getting \( k = 0 \) successes in 9 trials, given that the probability of success on a single trial is 0.21, is approximately:

\[
P(0) \approx 0.1199
\]

This means there is about a 11.99% chance of having no successes.

Would you like further details or clarifications?

Here are 5 related questions you might find helpful:
1. What is the probability of getting exactly 1 success in 9 trials with the same conditions?
2. How would the probability change if the probability of success was 0.5?
3. How do we calculate the expected number of successes in a binomial distribution?
4. What is the probability of getting at least one success in 9 trials?
5. How does the binomial distribution differ from a Poisson distribution?

**Tip:** The sum of probabilities for all possible outcomes (from \( k = 0 \) to \( k = n \)) in a binomial distribution always equals 1.

Learn how to calculate the probability of getting 0 successes in a binomial distribution where the probability of success is 0.21 and the number of trials is 9. This step-by-step solution covers key concepts in probability and the binomial theorem. Suitable for high school students in Grades 10-12.

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