If a coin is tossed till the first head appears, then what will be the sample space ?

Options

A. {H}
B. {TH}
C. {T, HT, HHT,HHHT, ……..}
D. {H, TH,TTH, TTTH, ……..}

Answer

To determine the sample space when tossing a coin until the first head appears, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Experiment**: We are tossing a coin repeatedly until we get the first head. This means we will keep tossing the coin until we see a head, and the experiment stops as soon as we get a head.

2. **Identifying Possible Outcomes**:
- The first possible outcome is getting a head on the first toss: **H**.
- If we get a tail on the first toss, we toss again. If we get a head on the second toss, the outcome is: **TH**.
- If we get tails on the first two tosses and then a head on the third toss, the outcome is: **TTH**.
- Continuing this pattern, if we get tails on the first three tosses and then a head on the fourth toss, the outcome is: **TTTH**.
- This pattern continues indefinitely.

3. **Generalizing the Outcomes**:
- The outcomes can be generalized as follows:
- If we get \( n \) tails followed by a head, the outcome can be represented as \( T^nH \), where \( n \) can be 0, 1, 2, 3, and so on.
- Therefore, the sample space can be expressed as:
- \( S = \{ H, TH, TTH, TTTH, TTTTH, \ldots \} \)

4. **Conclusion**: The sample space consists of all possible sequences of tails followed by the first head, which can be represented as:
- \( S = \{ H, TH, TTH, TTTH, TTTTH, \ldots \} \)

### Final Sample Space:
The sample space \( S \) is:
\[ S = \{ H, TH, TTH, TTTH, TTTTH, \ldots \} \]