An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

Options

A. `(63)/(64)`
B. `(255)/(256)`
C. `(127)/(128)`
D. `(1)/(2)`

Answer

To solve the problem of finding the probability of obtaining at least one head and at least one tail when tossing an unbiased coin eight times, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Outcomes**:
When a coin is tossed, there are 2 possible outcomes: heads (H) or tails (T). If the coin is tossed 8 times, the total number of possible outcomes is calculated as:
\[ \text{Total Outcomes} = 2^8 = 256 \]

2. **Calculate the Probability of All Heads**:
The probability of getting all heads in 8 tosses is:
\[ P(\text{All Heads}) = \frac{1}{256} \]

3. **Calculate the Probability of All Tails**:
Similarly, the probability of getting all tails in 8 tosses is:
\[ P(\text{All Tails}) = \frac{1}{256} \]

4. **Calculate the Probability of All Heads or All Tails**:
Since these two events (all heads and all tails) are mutually exclusive, we can add their probabilities:
\[ P(\text{All Heads or All Tails}) = P(\text{All Heads}) + P(\text{All Tails}) = \frac{1}{256} + \frac{1}{256} = \frac{2}{256} = \frac{1}{128} \]

5. **Calculate the Probability of At Least One Head and One Tail**:
To find the probability of getting at least one head and at least one tail, we can subtract the probability of getting all heads or all tails from 1:
\[ P(\text{At least one head and one tail}) = 1 - P(\text{All Heads or All Tails}) = 1 - \frac{1}{128} \]
\[ = \frac{128}{128} - \frac{1}{128} = \frac{127}{128} \]

### Final Answer:
The probability of obtaining at least one head and at least one tail when tossing an unbiased coin eight times is:
\[ \frac{127}{128} \]