mrs. gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students…

mrs. gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students. what is the probability that exactly 4 students have taken chemistry? round the answer to the nearest thousandth.\np(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}\n_nc_k=\frac{n!}{(n - k)!k!}\n0.005\n0.008\n0.213\n0.227

mrs. gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students. what is the probability that exactly 4 students have taken chemistry? round the answer to the nearest thousandth.\np(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}\n_nc_k=\frac{n!}{(n - k)!k!}\n0.005\n0.008\n0.213\n0.227

Answer

Explanation:

Step1: Identify values for formula

$n = 12$ (number of students surveyed), $k = 4$ (number of students who took chemistry), $p=0.4$ (probability a student takes chemistry).

Step2: Calculate combination ${n}C{k}$

${12}C{4}=\frac{12!}{(12 - 4)!4!}=\frac{12!}{8!4!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495$.

Step3: Calculate $p^{k}$ and $(1 - p)^{n - k}$

$p^{k}=(0.4)^{4}=0.0256$, $(1 - p)^{n - k}=(1 - 0.4)^{12 - 4}=(0.6)^{8}=0.01679616$.

Step4: Calculate probability $P(k)$

$P(4)={12}C{4}p^{4}(1 - p)^{12 - 4}=495\times0.0256\times0.01679616\approx0.213$.

Answer:

C. 0.213