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.\n\n$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$\n\n$_nc_k=\frac{n!}{(n - k)!k!}$\n\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.\n\n$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$\n\n$_nc_k=\frac{n!}{(n - k)!k!}$\n\n0.005\n0.008\n0.213\n0.227

Answer

Explanation:

Step1: Identify values of n, k, p

n = 12 (number of students surveyed), k = 4 (number of students taking chemistry), p = 0.4 (probability of a student taking 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}\times p^{4}\times(1 - p)^{8}=495\times0.0256\times0.01679616\approx0.213]

Answer:

0.213