Question

johnni guessed on all 8 questions of a multiple - choice quiz. each question has 4 answer choices. what is the probability that she got exactly 3 questions correct? round the answer to the nearest thousandth.
$p(k\text{ successes})={}_{n}c_{k}p^{k}(1 - p)^{n - k}$
${}_{n}c_{k}=\frac{n!}{(n - k)!k!}$
0.004
0.208
0.422
0.792

Explanation:

Step1: Identify values for formula

We have $n = 8$ (number of questions), $k=3$ (number of correct answers), and $p=\frac{1}{4}= 0.25$ (probability of getting a single - question correct since there are 4 choices per question).

Step2: Calculate the combination $_{n}C_{k}$

$_{8}C_{3}=\frac{8!}{(8 - 3)!3!}=\frac{8!}{5!3!}=\frac{8\times7\times6}{3\times2\times1}=56$

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

$p^{k}=(0.25)^{3}=0.25\times0.25\times0.25 = 0.015625$
$1-p=1 - 0.25=0.75$, and $(1 - p)^{n - k}=(0.75)^{5}=0.75\times0.75\times0.75\times0.75\times0.75\approx0.2373$

Step4: Calculate the probability $P(k)$

$P(3)=_{8}C_{3}p^{3}(1 - p)^{5}=56\times0.015625\times0.2373\approx0.208$

Answer:

0.208