To find the probability of no failures in five trials of a binomial experiment where the probability of success on a single trial is 30%, follow these steps:
1. Understand the Binomial Experiment Parameters:
- Number of trials, [tex]\( n \)[/tex]: 5
- Probability of success on each trial, [tex]\( p \)[/tex]: 30%, or 0.3
2. Define the Probability of Failure:
- The probability of failure on each trial, [tex]\( q \)[/tex], is given by [tex]\( 1 - p \)[/tex]:
[tex]\[
q = 1 - 0.3 = 0.7
\][/tex]
3. Identify the Number of Failures:
- We are looking for the probability of zero failures ([tex]\( k = 0 \)[/tex]) over the five trials.
4. Use the Binomial Probability Formula:
- The binomial probability formula is:
[tex]\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, calculated as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
5. Plug in the Values:
- For [tex]\( k = 0 \)[/tex]:
[tex]\[
\binom{5}{0} = \frac{5!}{0!(5-0)!} = \frac{1}{1 \cdot 1} = 1
\][/tex]
- So, the probability of zero failures is:
[tex]\[
P(X = 0) = 1 \cdot (0.3)^0 \cdot (0.7)^5
\][/tex]
- Simplifying:
[tex]\[
P(X = 0) = 1 \cdot 1 \cdot 0.16807 = 0.16807
\][/tex]
6. Convert the Probability to a Percentage:
- Multiply by 100 to convert the decimal probability to a percentage:
[tex]\[
0.16807 \times 100 = 16.807
\][/tex]
- Round to the nearest tenth of a percent:
[tex]\[
\text{Rounded Percentage} \approx 16.8\%
\][/tex]
Therefore, the probability of no failures in five trials of this binomial experiment, rounded to the nearest tenth of a percent, is:
[tex]\[
P \approx 16.8\%
\][/tex]
