Sure, let's break down this problem step-by-step.
To determine the probability of no failures in a binomial experiment with five trials, where the probability of success in each trial is given as 30%, we can follow these steps:
### Step 1: Understand the Parameters
- Number of Trials (n): 5
- Probability of Success in Each Trial (p): 0.30 (or 30%)
- Probability of Failure in Each Trial (q): 1 - p = 1 - 0.30 = 0.70 (or 70%)
### Step 2: Calculate the Probability of No Failures
In this context, "no failures" means that all five trials are successes. We can calculate this by raising the probability of success to the power of the number of trials.
[tex]\[ P(\text{no failures}) = (0.70)^5 \][/tex]
### Step 3: Compute the Probability
When we compute [tex]\((0.70)^5\)[/tex], we come up with the result:
[tex]\[ P(\text{no failures}) \approx 0.16807 \][/tex]
### Step 4: Express the Probability as a Percentage
To find the probability as a percentage, we multiply the result by 100:
[tex]\[ P(\text{no failures}) \approx 0.16807 \times 100 = 16.807\% \][/tex]
Thus, the probability of no failures in five trials, given that the probability of success in each trial is 30%, is approximately [tex]\(\boxed{16.807\%}\)[/tex].
