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 at most 4 students have taken chemistry? round the answer to the nearest thousandth.\n$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$\n$_nc_k=\frac{n!}{(n - k)!k!}$\n0 006\n0 438\n0 562\n0 665

mrs gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students. what is the probability that at most 4 students have taken chemistry? round the answer to the nearest thousandth.\n$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$\n$_nc_k=\frac{n!}{(n - k)!k!}$\n0 006\n0 438\n0 562\n0 665

Answer

Explanation:

Step1: Identify the binomial - distribution parameters

We have (n = 12) (number of trials/surveyed students), (p=0.4) (probability of success - student taking chemistry), and we want to find (P(X\leq4)). The binomial - probability formula is (P(X = k)={n}C{k}p^{k}(1 - p)^{n - k}), where ({n}C{k}=\frac{n!}{(n - k)!k!}).

Step2: Calculate (P(X = 0))

({12}C{0}=\frac{12!}{(12-0)!0!}=1), (p = 0.4), (1 - p=0.6), (k = 0). So (P(X = 0)={12}C{0}(0.4)^{0}(0.6)^{12}=1\times1\times(0.6)^{12}\approx0.0022).

Step3: Calculate (P(X = 1))

({12}C{1}=\frac{12!}{(12 - 1)!1!}=\frac{12!}{11!1!}=12), (P(X = 1)={12}C{1}(0.4)^{1}(0.6)^{11}=12\times0.4\times(0.6)^{11}\approx0.0174).

Step4: Calculate (P(X = 2))

({12}C{2}=\frac{12!}{(12 - 2)!2!}=\frac{12\times11}{2\times1}=66), (P(X = 2)={12}C{2}(0.4)^{2}(0.6)^{10}=66\times0.16\times(0.6)^{10}\approx0.0639).

Step5: Calculate (P(X = 3))

({12}C{3}=\frac{12!}{(12 - 3)!3!}=\frac{12\times11\times10}{3\times2\times1}=220), (P(X = 3)={12}C{3}(0.4)^{3}(0.6)^{9}=220\times0.064\times(0.6)^{9}\approx0.1419).

Step6: Calculate (P(X = 4))

({12}C{4}=\frac{12!}{(12 - 4)!4!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495), (P(X = 4)={12}C{4}(0.4)^{4}(0.6)^{8}=495\times0.0256\times(0.6)^{8}\approx0.2128).

Step7: Calculate (P(X\leq4))

(P(X\leq4)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)) (P(X\leq4)\approx0.0022 + 0.0174+0.0639+0.1419+0.2128\approx0.438).

Answer:

0.438