Certainly! To find the probability of exactly 4 successes in 7 trials of a binomial experiment where the probability of success is 55%, we need to use the binomial probability formula. Here is a step-by-step solution for the problem.
### Step-by-Step Solution
1. Identify the Parameters:
- Number of trials ([tex]\(n\)[/tex]): 7
- Probability of success in a single trial ([tex]\(p\)[/tex]): 0.55 (or 55% when expressed as a percentage)
- Number of successes ([tex]\(k\)[/tex]): 4
2. Understand the Binomial Probability Formula:
The binomial probability [tex]\(P(X = k)\)[/tex] of getting exactly [tex]\(k\)[/tex] successes in [tex]\(n\)[/tex] trials is given by:
[tex]\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
3. Calculate the Binomial Coefficient:
[tex]\[
\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\][/tex]
4. Substitute in the Binomial Formula:
[tex]\[
P(X = 4) = 35 \times (0.55)^4 \times (0.45)^3
\][/tex]
5. Calculate the Powers:
[tex]\[
(0.55)^4 \approx 0.0915078125
\][/tex]
[tex]\[
(0.45)^3 \approx 0.091125
\][/tex]
6. Combine to Find the Probability:
[tex]\[
P(X = 4) = 35 \times 0.0915078125 \times 0.091125 \approx 35 \times 0.008345275 = 0.2918477460937499
\][/tex]
7. Convert to Percentage:
[tex]\[
P(X = 4) \approx 0.2918477460937499 \times 100 \% \approx 29.2 \%
\][/tex]
### Final Answer
The probability of exactly 4 successes in 7 trials, with a success probability of 55%, rounded to the nearest tenth of a percent, is [tex]\(\boxed{29.2\%}\)[/tex].
