In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the com falls heads, he answers "true1; if it falls tails, he answers "false1. Find the probability that he gives at least two correct answers

Explanation

To solve the problem, we need to find the probability that a student gives at least two correct answers when answering 20 true-false questions by tossing a fair coin.

### Step-by-Step Solution:

1. **Define the Random Variable**:
Let \( X \) be the number of correct answers given by the student. Since the student answers each question based on a fair coin toss, \( X \) follows a binomial distribution.

2. **Parameters of the Binomial Distribution**:
The number of trials \( n = 20 \) (the number of questions), and the probability of success (getting a correct answer) for each question is \( p = \frac{1}{2} \) (since the coin is fair). The probability of failure (getting an incorrect answer) is also \( q = 1 - p = \frac{1}{2} \).

3. **Probability Mass Function**:
The probability of getting exactly \( r \) correct answers is given by the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
For our case, this becomes:
\[ P(X = r) = \binom{20}{r} \left(\frac{1}{2}\right)^r \left(\frac{1}{2}\right)^{20-r} = \binom{20}{r} \left(\frac{1}{2}\right)^{20} \]

4. **Finding the Probability of At Least Two Correct Answers**:
We need to find \( P(X \geq 2) \). This can be calculated using the complement rule:
\[ P(X \geq 2) = 1 - P(X < 2) = 1 - (P(X = 0) + P(X = 1)) \]

5. **Calculate \( P(X = 0) \)**:
\[ P(X = 0) = \binom{20}{0} \left(\frac{1}{2}\right)^{20} = 1 \cdot \left(\frac{1}{2}\right)^{20} = \frac{1}{2^{20}} \]

6. **Calculate \( P(X = 1) \)**:
\[ P(X = 1) = \binom{20}{1} \left(\frac{1}{2}\right)^{20} = 20 \cdot \left(\frac{1}{2}\right)^{20} = \frac{20}{2^{20}} \]

7. **Combine the Probabilities**:
\[ P(X < 2) = P(X = 0) + P(X = 1) = \frac{1}{2^{20}} + \frac{20}{2^{20}} = \frac{21}{2^{20}} \]

8. **Final Calculation**:
\[ P(X \geq 2) = 1 - P(X < 2) = 1 - \frac{21}{2^{20}} = \frac{2^{20} - 21}{2^{20}} \]

### Final Answer:
The probability that the student gives at least two correct answers is:
\[ P(X \geq 2) = \frac{2^{20} - 21}{2^{20}} \]