Sure, let's go through this step-by-step to solve the problem:
Given:
- The probability [tex]\( p \)[/tex] that a student has taken chemistry is [tex]\( 0.40 \)[/tex] (40%).
- Mrs. Gomes surveys [tex]\( n = 12 \)[/tex] students.
- We want to find the probability that exactly [tex]\( k = 4 \)[/tex] students have taken chemistry.
1. Understanding the Problem:
The problem asks us to find the probability of exactly 4 successes (students who have taken chemistry) out of 12 trials (students surveyed) when the probability of success in each trial is 0.40.
2. Using the Binomial Probability Formula:
The formula for the binomial probability of getting exactly [tex]\( k \)[/tex] successes in [tex]\( n \)[/tex] trials is given by:
[tex]\[
P(k \text{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
Where:
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, and it represents the number of ways to choose [tex]\( k \)[/tex] successes out of [tex]\( n \)[/tex] trials.
- [tex]\( p \)[/tex] is the probability of success on a single trial.
- [tex]\( (1-p) \)[/tex] is the probability of failure on a single trial.
3. Calculating the Binomial Coefficient:
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is found using:
[tex]\[
\binom{n}{k} = \frac{n!}{k! (n-k)!}
\][/tex]
For [tex]\( n = 12 \)[/tex] and [tex]\( k = 4 \)[/tex]:
[tex]\[
\binom{12}{4} = \frac{12!}{4! (12-4)!} = \frac{12!}{4! \cdot 8!}
\][/tex]
Using the given answer, the value of the binomial coefficient is 495.
4. Calculating the Probability:
Now, we use the binomial probability formula:
[tex]\[
P(4 \text{ successes}) = \binom{12}{4} \cdot (0.40)^4 \cdot (1-0.40)^{12-4}
\][/tex]
Substituting the values, we get:
[tex]\[
P(4 \text{ successes}) = 495 \cdot (0.40)^4 \cdot (0.60)^8
\][/tex]
Evaluating the expression, we find the probability to be approximately 0.21284093951999997.
5. Rounding the Answer:
The problem requires us to round the answer to the nearest thousandth:
[tex]\[
0.21284093951999997 \approx 0.213
\][/tex]
Conclusion:
The probability that exactly 4 students out of 12 have taken chemistry, rounded to the nearest thousandth, is [tex]\( \boxed{0.213} \)[/tex].
