To determine the probability that Van got exactly 1 question correct out of 8, when each question has 4 answer choices, we use the binomial probability formula:
[tex]\[
P(k \text{ successes }) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
Here, [tex]\( n \)[/tex] is the number of questions, [tex]\( k \)[/tex] is the number of successes (questions answered correctly), and [tex]\( p \)[/tex] is the probability of success for each question.
Let's break it down step by step:
1. Number of questions [tex]\( n \)[/tex]: 8
2. Number of successful guesses [tex]\( k \)[/tex]: 1
3. Probability of a successful guess [tex]\( p \)[/tex]: [tex]\(\frac{1}{4} = 0.25\)[/tex] (since there are 4 choices per question and only one correct choice).
First, we calculate the binomial coefficient [tex]\(\binom{n}{k}\)[/tex], which represents the number of ways to choose [tex]\( k \)[/tex] successes from [tex]\( n \)[/tex] trials:
[tex]\[
\binom{8}{1} = \frac{8!}{(8-1)!\cdot 1!} = \frac{8!}{7! \cdot 1!} = 8
\][/tex]
Next, we calculate the probability of exactly 1 success [tex]\( p^k \)[/tex]:
[tex]\[
p^k = (0.25)^1 = 0.25
\][/tex]
Then, we calculate the probability of [tex]\( n - k \)[/tex] failures:
[tex]\[
(1-p)^{n-k} = (1-0.25)^{8-1} = 0.75^7 \approx 0.133484
\][/tex]
Now, using these values, we calculate the overall probability using the binomial formula:
[tex]\[
P(1 \text{ success in 8 trials}) = \binom{8}{1} \cdot 0.25 \cdot 0.133484 = 8 \cdot 0.25 \cdot 0.133484 \approx 0.267
\][/tex]
Thus, the probability that Van got exactly 1 question correct, rounding to the nearest thousandth, is:
[tex]\[
\boxed{0.267}
\][/tex]
