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 for binomial formula

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: Substitute into binomial probability formula

$$ P(1)=8\times(0.25)^1\times(1-0.25)^{8-1} $$

Step4: Compute each term

$(0.25)^1=0.25$, $(0.75)^7\approx0.13348$

$$ P(1)=8\times0.25\times0.13348\approx0.267 $$

Answer:

0.267