One coin is thrown 100 times. The probability of coming tails in odd number

Options

A. `1/2`
B. `1/8`
C. `3/8`
D. `3/4`

Answer

To solve the problem of finding the probability of getting tails an odd number of times when a coin is thrown 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 when a coin is tossed 100 times.

2. **Defining the Events**:
Let \( X \) be the random variable representing the number of tails obtained in 100 tosses. We are interested in \( P(X \text{ is odd}) \).

3. **Using Binomial Distribution**:
The number of tails in 100 tosses follows a binomial distribution \( B(n, p) \) where \( n = 100 \) and \( p = \frac{1}{2} \) (the probability of getting tails in a single toss).

4. **Calculating the Total Probability**:
The total probability of getting tails an odd number of times can be calculated as:
\[ P(X \text{ is odd}) = P(X = 1) + P(X = 3) + P(X = 5) + \ldots + P(X = 99) \]
Each of these probabilities can be calculated using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
For our case, this becomes:
\[ P(X = k) = \binom{100}{k} \left(\frac{1}{2}\right)^{100} \]

5. **Summing the Probabilities**:
We can factor out \( \left(\frac{1}{2}\right)^{100} \):
\[ P(X \text{ is odd}) = \left(\frac{1}{2}\right)^{100} \left( \sum_{k \text{ odd}} \binom{100}{k} \right) \]

6. **Using the Binomial Theorem**:
The sum of the binomial coefficients for odd \( k \) can be found using the identity:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
and
\[ \sum_{k \text{ odd}} \binom{n}{k} = \frac{1}{2} \left( 2^n - (1)^n \right) \]
For \( n = 100 \):
\[ \sum_{k \text{ odd}} \binom{100}{k} = \frac{1}{2} \left( 2^{100} - 1 \right) \]

7. **Final Calculation**:
Therefore,
\[ P(X \text{ is odd}) = \left(\frac{1}{2}\right)^{100} \cdot \frac{1}{2} \left( 2^{100} - 1 \right) = \frac{1}{2} \left( 1 - \frac{1}{2^{100}} \right) \]
As \( \frac{1}{2^{100}} \) is negligible, we can approximate:
\[ P(X \text{ is odd}) \approx \frac{1}{2} \]

8. **Conclusion**:
Thus, the probability of getting tails an odd number of times when a coin is thrown 100 times is:
\[ P(X \text{ is odd}) = \frac{1}{2} \]