Question

johnny 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.

$p(k\text{ successes}) = {_nc_k}p^k(1 - p)^{n - k}$

${_nc_k}=\frac{n!}{(n - k)!k!}$

\bigcirc 0.004
\bigcirc 0.208
\bigcirc 0.422
\bigcirc 0.792

Explanation:

Step1: Identify parameters

\( n = 8 \) (questions), \( k = 3 \) (correct), \( p=\frac{1}{4}=0.25 \) (prob correct), \( 1 - p = 0.75 \) (prob wrong).

Step2: Calculate combination \( _nC_k \)

\( _8C_3=\frac{8!}{(8 - 3)!3!}=\frac{8!}{5!3!}=\frac{8\times7\times6}{3\times2\times1}=56 \)

Step3: Apply binomial formula

\( P(3)=\ _8C_3\times(0.25)^3\times(0.75)^{8 - 3} \)
\( = 56\times0.015625\times0.2373046875 \)
\( = 56\times0.0037078125 \)
\( \approx0.20764375 \approx 0.208 \)

Answer:

0.208 (corresponding to the option "0.208")