Certainly! Let's go through a detailed, step-by-step solution to determine the probability that Stan got at least 2 questions correct on a 10-question multiple-choice quiz where each question has 4 answer choices.
### Step-by-Step Solution
1. Identify the Given Values:
- Number of questions ([tex]\( n \)[/tex]): 10
- Probability of guessing a question correctly ([tex]\( p \)[/tex]): Since each question has 4 answer choices, [tex]\( p = \frac{1}{4} = 0.25 \)[/tex].
- Probability of guessing a question incorrectly ([tex]\( q \)[/tex]): [tex]\( q = 1 - p = 1 - 0.25 = 0.75 \)[/tex].
- We are interested in calculating the probability of getting at least 2 questions correct.
2. Binomial Probability Formula:
The probability of getting exactly [tex]\( k \)[/tex] correct answers out of [tex]\( n \)[/tex] questions is given by the binomial probability formula:
[tex]\[
P(k \text{ correct}) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] (read as "n choose k") is the combination of [tex]\( n \)[/tex] items taken [tex]\( k \)[/tex] at a time.
3. Compute the Individual Probabilities:
First, we need to compute the probability for each possible number of correct answers from 2 to 10. This involves calculating [tex]\(\binom{10}{k}\)[/tex], using the formula:
[tex]\[
\binom{10}{k} = \frac{10!}{k! \cdot (10-k)!}
\][/tex]
Next, we use the binomial formula to compute the probability for each [tex]\( k \)[/tex]:
[tex]\[
P(k \text{ correct}) = \binom{10}{k} (0.25)^k (0.75)^{10-k}
\][/tex]
We need to calculate this for [tex]\( k = 2 \)[/tex] to [tex]\( k = 10 \)[/tex].
4. Sum of Probabilities for At Least 2 Correct Answers:
The probability of getting at least 2 correct answers is the sum of the probabilities of getting 2, 3, ..., 10 correct answers:
[tex]\[
P(\text{at least 2 correct}) = \sum_{k=2}^{10} P(k \text{ correct})
\][/tex]
5. Result:
After summing up all these probabilities, we get the final result. The hard work of summing all combinations and probabilities provides the cumulative probability.
### Final Probability
The probability that Stan got at least 2 questions correct on his 10-question multiple-choice quiz, rounding to the nearest thousandth, is:
[tex]\[ \boxed{0.756} \][/tex]
In conclusion, Stan has a 0.756 probability, or 75.6% chance, of guessing at least 2 out of 10 questions correctly.
