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.
$p(k\text{ successes})=_{n}c_{k}p^{k}(1-p)^{n-k}$
$_{n}c_{k}=\frac{n!}{(n-k)!\cdot k!}$
0.033
0.267
0.461
0.733

Explanation:

Step1: Define variables

Let $n=8$ (total questions), $k=1$ (correct answers), $p=\frac{1}{4}=0.25$ (probability of correct guess).

Step2: Calculate combination $_nC_k$

$$_8C_1=\frac{8!}{(8-1)!\cdot1!}=\frac{8!}{7!\cdot1!}=8$$

Step3: Compute binomial probability

$$P(1)=_8C_1p^1(1-p)^{8-1}=8\times(0.25)^1\times(0.75)^7$$
First calculate $(0.75)^7\approx0.13348$, then:
$$P(1)=8\times0.25\times0.13348=2\times0.13348=0.26696$$

Step4: Round to nearest thousandth

$0.26696\approx0.267$

Answer:

0.267