Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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].
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].
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.