Answer :
Let's solve this problem step by step.
1. Understand the problem:
- Stan answered 10 questions.
- Each question has 4 possible answers, giving us a probability \( p = \frac{1}{4} \) or 0.25 for selecting the correct answer.
- We are asked to find the probability that Stan got at least 2 questions correct.
- We will use the binomial probability formula to calculate this.
2. Define the probabilities:
- Let \( n = 10 \) be the total number of questions.
- Let \( p = 0.25 \) be the probability of guessing one question correctly.
- Let’s denote the probability of getting exactly \( k \) questions correct by \( P(X = k) \).
3. Calculate the cumulative probability for getting 0 or 1 questions correct:
- For \( k = 0 \):
[tex]\[ P(X = 0) = \binom{10}{0} p^0 (1-p)^{10} = 1 \cdot (0.25)^0 \cdot (0.75)^{10} = 1 \cdot 1 \cdot 0.0563 \approx 0.056 \][/tex]
- For \( k = 1 \):
[tex]\[ P(X = 1) = \binom{10}{1} p^1 (1-p)^{9} = 10 \cdot (0.25)^1 \cdot (0.75)^9 = 10 \cdot 0.25 \cdot 0.1937 \approx 0.193 \][/tex]
4. Sum the probabilities for getting 0 or 1 questions correct:
[tex]\[ P(X = 0) + P(X = 1) = 0.056 + 0.193 = 0.244 \][/tex]
5. Calculate the probability of getting at least 2 questions correct:
- The probability of getting at least 2 questions correct is the complement of getting 0 or 1 questions correct.
[tex]\[ P(\text{at least 2 correct}) = 1 - P(X \leq 1) = 1 - 0.244 = 0.756 \][/tex]
6. Round the result to the nearest thousandth:
- The probability that Stan got at least 2 questions correct is approximately 0.756 when rounded to the nearest thousandth.
So, the probability that Stan got at least 2 questions correct is [tex]\( \boxed{0.756} \)[/tex].
1. Understand the problem:
- Stan answered 10 questions.
- Each question has 4 possible answers, giving us a probability \( p = \frac{1}{4} \) or 0.25 for selecting the correct answer.
- We are asked to find the probability that Stan got at least 2 questions correct.
- We will use the binomial probability formula to calculate this.
2. Define the probabilities:
- Let \( n = 10 \) be the total number of questions.
- Let \( p = 0.25 \) be the probability of guessing one question correctly.
- Let’s denote the probability of getting exactly \( k \) questions correct by \( P(X = k) \).
3. Calculate the cumulative probability for getting 0 or 1 questions correct:
- For \( k = 0 \):
[tex]\[ P(X = 0) = \binom{10}{0} p^0 (1-p)^{10} = 1 \cdot (0.25)^0 \cdot (0.75)^{10} = 1 \cdot 1 \cdot 0.0563 \approx 0.056 \][/tex]
- For \( k = 1 \):
[tex]\[ P(X = 1) = \binom{10}{1} p^1 (1-p)^{9} = 10 \cdot (0.25)^1 \cdot (0.75)^9 = 10 \cdot 0.25 \cdot 0.1937 \approx 0.193 \][/tex]
4. Sum the probabilities for getting 0 or 1 questions correct:
[tex]\[ P(X = 0) + P(X = 1) = 0.056 + 0.193 = 0.244 \][/tex]
5. Calculate the probability of getting at least 2 questions correct:
- The probability of getting at least 2 questions correct is the complement of getting 0 or 1 questions correct.
[tex]\[ P(\text{at least 2 correct}) = 1 - P(X \leq 1) = 1 - 0.244 = 0.756 \][/tex]
6. Round the result to the nearest thousandth:
- The probability that Stan got at least 2 questions correct is approximately 0.756 when rounded to the nearest thousandth.
So, the probability that Stan got at least 2 questions correct is [tex]\( \boxed{0.756} \)[/tex].
