Sure, let's solve this step by step.
The problem involves finding the probability of [tex]\(x\)[/tex] successes in [tex]\(n\)[/tex] trials with a success probability of [tex]\(p\)[/tex] for each trial. This is a typical binomial probability problem. Let's break it down.
### Given:
- Number of trials ([tex]\(n\)[/tex]) = 8
- Number of successes ([tex]\(x\)[/tex]) = 0
- Probability of success in each trial ([tex]\(p\)[/tex]) = 0.65
- Probability of failure in each trial ([tex]\(q = 1 - p\)[/tex]) = [tex]\(1 - 0.65 = 0.35\)[/tex]
### Step-by-Step Solution:
1. Binomial Coefficient ([tex]\( \binom{n}{x} \)[/tex]):
[tex]\[
\binom{n}{x} = \frac{n!}{x!(n-x)!}
\][/tex]
For [tex]\( n = 8 \)[/tex] and [tex]\( x = 0 \)[/tex]:
[tex]\[
\binom{8}{0} = \frac{8!}{0!(8-0)!} = \frac{8!}{0! \cdot 8!} = 1
\][/tex]
2. Calculating the Success Probability:
[tex]\[
p^x
\][/tex]
Since the number of successes [tex]\( x = 0 \)[/tex]:
[tex]\[
0.65^0 = 1
\][/tex]
3. Calculating the Failure Probability:
[tex]\[
(1 - p)^{n-x}
\][/tex]
Since [tex]\( 1 - p = 0.35 \)[/tex] and [tex]\( n-x = 8-0 = 8 \)[/tex]:
[tex]\[
0.35^8 = 0.0002251875390624999
\][/tex]
4. Multiplying These Values Together:
[tex]\[
\binom{8}{0} \times 0.65^0 \times 0.35^8
\][/tex]
Substituting the values we calculated:
[tex]\[
1 \times 1 \times 0.0002251875390624999 = 0.0002251875390624999
\][/tex]
Thus, the probability of getting exactly 0 successes out of 8 trials when the probability of success in each trial is 0.65 is [tex]\( 0.0002251875390624999 \)[/tex].
