Question

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.
p(x = k)={}_{n}c_{k}p^{k}(1 - p)^{n - k}
{}_{n}c_{k}=\frac{n!}{(n - k)!k!}
0.013
0.267
0.451
0.733

Explanation:

Step1: Identify values for binomial formula

We use the binomial probability formula $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$. Here, $n$ (number of trials) is the number of questions $n = 8$, $k$ (number of successes) is the number of correct - answers $k = 1$, and $p$ (probability of success on a single trial) is the probability of getting a single question correct. Since each question has 4 answer choices, $p=\frac{1}{4}=0.25$ and $1 - p = 0.75$. The combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Calculate the combination $C(8,1)$

$C(8,1)=\frac{8!}{1!(8 - 1)!}=\frac{8!}{1!7!}=\frac{8\times7!}{1\times7!}=8$.

Step3: Calculate the binomial probability

$P(X = 1)=C(8,1)\times(0.25)^{1}\times(0.75)^{8 - 1}$.
$P(X = 1)=8\times0.25\times(0.75)^{7}$.
$(0.75)^{7}=0.75\times0.75\times0.75\times0.75\times0.75\times0.75\times0.75\approx0.1335$.
$8\times0.25\times0.1335 = 0.267$.

Answer:

$0.267$