van guessed on all 8 questions of a multiple - choice quiz. each question has 4 answer choices. what is the…

van guessed on all 8 questions of a multiple - choice quiz. each question has 4 answer choices. what is the probability that he got exactly 1 question correct? 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!}\n0.033\n0.267\n0.461\n0.733

van guessed on all 8 questions of a multiple - choice quiz. each question has 4 answer choices. what is the probability that he got exactly 1 question correct? 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!}\n0.033\n0.267\n0.461\n0.733

Answer

Explanation:

Step1: Identify values of n, k, p

n = 8 (number of questions), k = 1 (number of correct answers), p = 1/4 (probability of getting a single - question correct)

Step2: Calculate combination ({n}C{k})

({n}C{k}=\frac{n!}{(n - k)!k!}), so ({8}C{1}=\frac{8!}{(8 - 1)!1!}=\frac{8!}{7!1!}=\frac{8\times7!}{7!×1}=8)

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

(p=\frac{1}{4}), (1-p = 1-\frac{1}{4}=\frac{3}{4}), (k = 1), (n = 8) (p^{k}(1 - p)^{n - k}=(\frac{1}{4})^{1}\times(\frac{3}{4})^{8 - 1}=\frac{1}{4}\times(\frac{3}{4})^{7}=\frac{1}{4}\times\frac{2187}{16384}=\frac{2187}{65536})

Step4: Calculate (P(k))

(P(k)={n}C{k}p^{k}(1 - p)^{n - k}) (P(1)=8\times\frac{2187}{65536}=\frac{17496}{65536}\approx0.267)

Answer:

0.267