Question

stan guessed on all 10 questions of a multiple-choice quiz. each question has 4 answer choices. what is the probability that he got at least 2 questions correct? round the answer to the nearest thousandth.
p(k successes) = ( _nc_k p^k (1 - p)^{n - k} )
( _nc_k = \frac{n!}{(n - k)! cdot k!} )
○ 0.211
○ 0.244
○ 0.756
○ 0.944

Explanation:

Step1: Identify Distribution

This is a binomial probability problem. \( n = 10 \) (questions), \( p=\frac{1}{4}=0.25 \) (probability of correct guess), \( 1 - p = 0.75 \). We need \( P(X\geq2)=1 - P(X = 0)-P(X = 1) \).

Step2: Calculate \( P(X = 0) \)

Using \( P(k)=\binom{n}{k}p^{k}(1 - p)^{n - k} \), for \( k = 0 \):
\( \binom{10}{0}=\frac{10!}{0!10!}=1 \), \( P(0)=1\times(0.25)^{0}\times(0.75)^{10}=0.75^{10}\approx0.0563 \)

Step3: Calculate \( P(X = 1) \)

For \( k = 1 \):
\( \binom{10}{1}=\frac{10!}{1!9!}=10 \), \( P(1)=10\times(0.25)^{1}\times(0.75)^{9}=10\times0.25\times0.75^{9}\approx10\times0.25\times0.0751\approx0.1877 \)

Step4: Calculate \( P(X\geq2) \)

\( P(X\geq2)=1 - 0.0563 - 0.1877 = 1 - 0.244 = 0.756 \)

Answer:

0.756 (corresponding to the option with 0.756)