To determine the probability that Van guessed exactly 1 question correctly on an 8-question multiple-choice quiz, we can use the binomial probability formula. Here are the steps involved:
### Step 1: Identify the Given Values
- Total number of questions ([tex]\(n\)[/tex]): 8
- Number of correct answers ([tex]\(k\)[/tex]): 1
- Probability of guessing one question correctly ([tex]\(p\)[/tex]): [tex]\(\frac{1}{4}\)[/tex] (since there are 4 choices per question)
### Step 2: Probability of Guessing Incorrectly
- Since the probability of guessing correctly is [tex]\(\frac{1}{4}\)[/tex], the probability of guessing incorrectly ([tex]\(q\)[/tex]) is:
[tex]\[
q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}
\][/tex]
### Step 3: Calculate the Combination [tex]\({ }_n C_k\)[/tex]
[tex]\[
{ }_n C _k = \frac{n!}{(n-k)!\cdot k!}
\][/tex]
Plugging in the values:
[tex]\[
{ }_8 C _1 = \frac{8!}{(8-1)!\cdot 1!} = \frac{8!}{7! \cdot 1!} = \frac{8 \cdot 7!}{7! \cdot 1} = 8
\][/tex]
### Step 4: Apply the Binomial Probability Formula
[tex]\[
P(k \text{ successes}) = { }_n C_k \cdot p^k \cdot (1-p)^{n-k}
\][/tex]
Substitute the values:
[tex]\[
P(1 \text{ success}) = 8 \cdot \left(\frac{1}{4}\right)^1 \cdot \left(\frac{3}{4}\right)^7
\][/tex]
### Step 5: Calculate the Probability
First, calculate [tex]\(\left(\frac{1}{4}\right)^1\)[/tex]:
[tex]\[
\left(\frac{1}{4}\right)^1 = \frac{1}{4}
\][/tex]
Next, calculate [tex]\(\left(\frac{3}{4}\right)^7\)[/tex]:
[tex]\[
\left(\frac{3}{4}\right)^7 \approx 0.1335
\][/tex]
Then, multiply everything together:
[tex]\[
P(1 \text{ success}) = 8 \cdot \frac{1}{4} \cdot 0.1335 = 8 \cdot 0.25 \cdot 0.1335 = 0.267
\][/tex]
### Step 6: Round to the Nearest Thousandth
[tex]\[
\text{Rounded Probability} = 0.267
\][/tex]
### Conclusion
The probability that Van got exactly 1 question correct is approximately [tex]\(0.267\)[/tex].
So the correct answer is:
[tex]\[
\boxed{0.267}
\][/tex]
