To find the probability of exactly three successes in seven trials of a binomial experiment where the probability of success is 35%, we can use the binomial probability formula. The binomial probability formula is given by:
[tex]\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where:
- [tex]\( n \)[/tex] is the number of trials,
- [tex]\( k \)[/tex] is the number of successes,
- [tex]\( p \)[/tex] is the probability of success on a single trial,
- [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, which can be calculated as [tex]\( \frac{n!}{k!(n-k)!} \)[/tex].
Given:
- [tex]\( n = 7 \)[/tex]
- [tex]\( k = 3 \)[/tex]
- [tex]\( p = 0.35 \)[/tex]
1. Calculate the binomial coefficient [tex]\( \binom{7}{3} \)[/tex]:
[tex]\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\][/tex]
2. Calculate [tex]\( p^k \)[/tex]:
[tex]\[
p^3 = (0.35)^3 = 0.042875
\][/tex]
3. Calculate [tex]\( (1-p)^{n-k} \)[/tex]:
[tex]\[
(0.65)^4 = 0.17850625
\][/tex]
4. Combine these values into the binomial probability formula:
[tex]\[
P(X = 3) = 35 \times 0.042875 \times 0.17850625 \approx 0.26787094140625
\][/tex]
5. Convert the probability to a percentage:
[tex]\[
0.26787094140625 \times 100 \approx 26.787094140625
\][/tex]
6. Round this to the nearest tenth of a percent:
[tex]\[
26.787094140625 \approx 26.8
\][/tex]
Therefore, the probability of having exactly three successes in seven trials, with a success probability of 35% per trial, is approximately [tex]\( 26.8 \% \)[/tex].
