The probability of guessing atleast 8 correct answers out of 10 true /false questions

Options

A. `7/64`
B. `7/128`
C. `7/256`
D. `35/(1024)`

Answer

To solve the problem of finding the probability of guessing at least 8 correct answers out of 10 true/false questions, we can follow these steps:

### Step 1: Define the Probability of Correct and Incorrect Answers
The probability of guessing a correct answer (True or False) is \( \frac{1}{2} \) and the probability of guessing an incorrect answer is also \( \frac{1}{2} \).

**Hint:** Remember that each question has two possible outcomes, hence the equal probabilities.

### Step 2: Identify the Required Outcomes
We need to calculate the probability of getting at least 8 correct answers. This includes:
- Getting exactly 8 correct answers
- Getting exactly 9 correct answers
- Getting exactly 10 correct answers

**Hint:** "At least 8" means we need to consider 8, 9, and 10 correct answers.

### Step 3: Use the Binomial Probability Formula
The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:
- \( n \) is the total number of trials (questions),
- \( k \) is the number of successful trials (correct answers),
- \( p \) is the probability of success on an individual trial.

For our case, \( n = 10 \) and \( p = \frac{1}{2} \).

### Step 4: Calculate the Probabilities
1. **For 8 correct answers:**
\[ P(X = 8) = \binom{10}{8} \left(\frac{1}{2}\right)^8 \left(\frac{1}{2}\right)^{2} = \binom{10}{8} \left(\frac{1}{2}\right)^{10} \]
\[ = \binom{10}{2} \left(\frac{1}{2}\right)^{10} \]
\[ = 45 \cdot \left(\frac{1}{2}\right)^{10} \]

2. **For 9 correct answers:**
\[ P(X = 9) = \binom{10}{9} \left(\frac{1}{2}\right)^9 \left(\frac{1}{2}\right)^{1} = \binom{10}{9} \left(\frac{1}{2}\right)^{10} \]
\[ = 10 \cdot \left(\frac{1}{2}\right)^{10} \]

3. **For 10 correct answers:**
\[ P(X = 10) = \binom{10}{10} \left(\frac{1}{2}\right)^{10} = 1 \cdot \left(\frac{1}{2}\right)^{10} \]

### Step 5: Combine the Probabilities
Now, we combine the probabilities of getting 8, 9, and 10 correct answers:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) \]
\[ = 45 \cdot \left(\frac{1}{2}\right)^{10} + 10 \cdot \left(\frac{1}{2}\right)^{10} + 1 \cdot \left(\frac{1}{2}\right)^{10} \]
\[ = (45 + 10 + 1) \cdot \left(\frac{1}{2}\right)^{10} = 56 \cdot \left(\frac{1}{2}\right)^{10} \]

### Step 6: Simplify the Expression
\[ P(X \geq 8) = \frac{56}{1024} = \frac{7}{128} \]

### Final Answer
The probability of guessing at least 8 correct answers out of 10 true/false questions is \( \frac{7}{128} \).

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