To determine the probability that Van guessed exactly 1 question correctly out of 8 questions, where each question has 4 answer choices, we can use the binomial probability formula.
The formula for binomial probability is:
[tex]\[ P(k \text{ successes}) = _nC_k \cdot p^k \cdot (1-p)^{n-k} \][/tex]
where:
- [tex]\( n = 8 \)[/tex] (the total number of questions),
- [tex]\( k = 1 \)[/tex] (the number of correct questions we are interested in),
- [tex]\( p = \frac{1}{4} \)[/tex] (the probability of guessing a question correctly, since each question has 4 choices, only one of which is correct),
- [tex]\( 1-p = \frac{3}{4} \)[/tex] (the probability of guessing a question incorrectly).
1. Calculate the binomial coefficient [tex]\(_nC_k\)[/tex]:
The binomial coefficient [tex]\(_nC_k\)[/tex] is calculated as:
[tex]\[ _nC_k = \frac{n!}{k!(n-k)!} \][/tex]
Plugging in the values:
[tex]\[ _8C_1 = \frac{8!}{1!(8-1)!} = \frac{8!}{1!7!} = \frac{8 \cdot 7!}{1! \cdot 7!} = 8 \][/tex]
2. Calculate the probability using the binomial formula:
[tex]\[ P(1 \text{ success}) = _8C_1 \cdot p^1 \cdot (1-p)^{8-1} \][/tex]
[tex]\[ P(1 \text{ success}) = 8 \cdot \left(\frac{1}{4}\right)^1 \cdot \left(\frac{3}{4}\right)^7 \][/tex]
3. Evaluate the expression step-by-step:
[tex]\[ P(1 \text{ success}) = 8 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^7 \][/tex]
[tex]\[ P(1 \text{ success}) = 8 \cdot \frac{1}{4} \cdot \left(\frac{2187}{16384}\right) \quad \text{(since } \left(\frac{3}{4}\right)^7 = \frac{3^7}{4^7} = \frac{2187}{16384} \text{)} \][/tex]
[tex]\[ P(1 \text{ success}) = 8 \cdot \frac{1}{4} \cdot \frac{2187}{16384} = 2 \cdot \frac{2187}{16384} = \frac{4374}{16384} \][/tex]
[tex]\[ P(1 \text{ success}) \approx 0.2669677734375 \][/tex]
4. Round the answer to the nearest thousandth:
[tex]\[ P(1 \text{ success}) \approx 0.267 \][/tex]
Thus, the probability that Van guessed exactly 1 question correctly is approximately [tex]\( \boxed{0.267} \)[/tex].
