To find the probability that exactly 4 out of 12 randomly surveyed students have taken chemistry, we'll use the binomial distribution formula. Here, the probability of success ([tex]\( p \)[/tex]) is 0.4 (since 40% of students take chemistry), the total number of trials ([tex]\( n \)[/tex]) is 12, and the number of successes ([tex]\( k \)[/tex]) we are interested in is 4.
The formula for the binomial probability is given by:
[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \][/tex]
Where:
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex]
- [tex]\(n\)[/tex] is the total number of trials (students surveyed)
- [tex]\(k\)[/tex] is the number of successes (students who have taken chemistry)
- [tex]\(p\)[/tex] is the probability of success on a single trial
Step-by-Step Solution:
1. Calculate the binomial coefficient [tex]\(\binom{n}{k}\)[/tex]:
[tex]\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} = 495 \][/tex]
2. Calculate [tex]\( p^k \)[/tex]:
[tex]\[ 0.4^4 = 0.4 \cdot 0.4 \cdot 0.4 \cdot 0.4 = 0.0256 \][/tex]
3. Calculate [tex]\( (1-p)^{n-k} \)[/tex]:
[tex]\[ (1-0.4)^{12-4} = 0.6^8 = 0.1679616 \][/tex]
4. Combine these values into the binomial probability formula:
[tex]\[ P(X = 4) = 495 \cdot 0.0256 \cdot 0.1679616 \][/tex]
5. Calculate the probability:
[tex]\[ P(X = 4) = 495 \cdot 0.0256 \cdot 0.1679616 = 495 \cdot 0.0043006592 = 0.21284093952 \][/tex]
6. Round the probability to the nearest thousandth:
[tex]\[ P(X = 4) \approx 0.213 \][/tex]
Therefore, the probability that exactly 4 out of 12 students have taken chemistry is approximately [tex]\( 0.213 \)[/tex]. The correct answer is [tex]\( 0.213 \)[/tex].
