Answer :
To solve the problem of finding the probability of no successes in eight trials of a binomial experiment where the probability of success is \(7\%\) (or 0.07), we can use the formula for the binomial distribution. Specifically, we want the probability of zero successes (\(k = 0\)).
The formula for the probability of exactly \(k\) successes in \(n\) trials of a binomial experiment, where the probability of success on a single trial is \(p\), is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \][/tex]
For this problem:
- \(n = 8\) (the number of trials)
- \(k = 0\) (we are looking for the probability of zero successes)
- \(p = 0.07\) (the probability of success on a single trial)
Substituting these values into our formula, we get:
[tex]\[ P(X = 0) = \binom{8}{0} (0.07)^0 (1 - 0.07)^{8 - 0} \][/tex]
\(\binom{8}{0}\) represents the number of ways to choose 0 successes out of 8 trials, which is 1, and \((0.07)^0\) is also 1. This simplifies our equation to:
[tex]\[ P(X = 0) = 1 \cdot 1 \cdot (1 - 0.07)^8 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ 1 - 0.07 = 0.93 \][/tex]
Then, raise 0.93 to the power of 8:
[tex]\[ (0.93)^8 \][/tex]
Calculating this expression gives the probability in decimal form. Finally, to convert this probability to a percentage, multiply by 100.
[tex]\[ P(X = 0) = (0.93)^8 \approx 0.5596 \][/tex]
Converting to a percentage:
[tex]\[ 0.5596 \times 100 \approx 55.96\% \][/tex]
Therefore, the probability of no successes in eight trials of a binomial experiment where the probability of success on each trial is \(7\%\) is approximately \(55.96\%\).
Thus,
[tex]\[ P = 55.96\% \][/tex]
The formula for the probability of exactly \(k\) successes in \(n\) trials of a binomial experiment, where the probability of success on a single trial is \(p\), is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \][/tex]
For this problem:
- \(n = 8\) (the number of trials)
- \(k = 0\) (we are looking for the probability of zero successes)
- \(p = 0.07\) (the probability of success on a single trial)
Substituting these values into our formula, we get:
[tex]\[ P(X = 0) = \binom{8}{0} (0.07)^0 (1 - 0.07)^{8 - 0} \][/tex]
\(\binom{8}{0}\) represents the number of ways to choose 0 successes out of 8 trials, which is 1, and \((0.07)^0\) is also 1. This simplifies our equation to:
[tex]\[ P(X = 0) = 1 \cdot 1 \cdot (1 - 0.07)^8 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ 1 - 0.07 = 0.93 \][/tex]
Then, raise 0.93 to the power of 8:
[tex]\[ (0.93)^8 \][/tex]
Calculating this expression gives the probability in decimal form. Finally, to convert this probability to a percentage, multiply by 100.
[tex]\[ P(X = 0) = (0.93)^8 \approx 0.5596 \][/tex]
Converting to a percentage:
[tex]\[ 0.5596 \times 100 \approx 55.96\% \][/tex]
Therefore, the probability of no successes in eight trials of a binomial experiment where the probability of success on each trial is \(7\%\) is approximately \(55.96\%\).
Thus,
[tex]\[ P = 55.96\% \][/tex]
