A fair coin is tossed 100 times. The probability of getting tails an odd number of times is `1//2` b. `1//8` c. `3//8` d. none of these

Options

A. `1//2`
B. `1//8`
C. `3//8`
D. none of these

Explanation

To solve the problem of finding the probability of getting tails an odd number of times when a fair coin is tossed 100 times, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find the probability of getting tails an odd number of times in 100 tosses of a fair coin.

2. **Defining Events**:
- Let \( X \) be the random variable representing the number of tails obtained in 100 tosses.
- Since the coin is fair, the probability of getting tails (success) in a single toss is \( p = \frac{1}{2} \) and the probability of not getting tails (failure) is \( q = 1 - p = \frac{1}{2} \).

3. **Using Binomial Distribution**:
- The number of tails in 100 tosses follows a binomial distribution: \( X \sim \text{Binomial}(n=100, p=\frac{1}{2}) \).
- The probability mass function of a binomial distribution is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
- For our case, \( n = 100 \), \( p = \frac{1}{2} \), and \( q = \frac{1}{2} \).

4. **Calculating the Probability of Odd Tails**:
- We want to find the probability of getting tails an odd number of times:
\[ P(X \text{ is odd}) = P(X = 1) + P(X = 3) + P(X = 5) + \ldots + P(X = 99) \]

5. **Using the Binomial Theorem**:
- The sum of probabilities for odd \( k \) can be simplified using the binomial theorem:
\[ P(X \text{ is odd}) = \frac{1}{2} \left( P(X = 0) + P(X = 1) + P(X = 2) + \ldots + P(X = 100) \right) \]
- The total probability \( P(X = 0) + P(X = 1) + \ldots + P(X = 100) = 1 \) (since it covers all possible outcomes).

6. **Final Calculation**:
- Therefore, we have:
\[ P(X \text{ is odd}) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Conclusion:
The probability of getting tails an odd number of times when a fair coin is tossed 100 times is \( \frac{1}{2} \).