If a coin is tossed till the first head appears, then what will be the sample space?
{H, TH, TTH, TTTH, ……….}
Understanding the Sample Space of a Coin Toss
The question asks for the sample space when a coin is tossed repeatedly until the first head appears. The sample space of a random experiment is the set of all possible outcomes.
Defining the Experiment
The experiment stops as soon as a head (H) is observed. This means we keep tossing the coin if we get a tail (T), and we stop as soon as we get a head.
Identifying Possible Outcomes
Let's think about the possible sequence of outcomes in this experiment:
- Outcome 1: The very first toss is a head. The experiment stops immediately. The outcome is H.
- Outcome 2: The first toss is a tail (T), and the second toss is a head (H). The experiment stops. The outcome is TH.
- Outcome 3: The first two tosses are tails (T), and the third toss is a head (H). The experiment stops. The outcome is TTH.
- Outcome 4: The first three tosses are tails (T), and the fourth toss is a head (H). The experiment stops. The outcome is TTTH.
This process can continue indefinitely. We could have any number of tails followed by a single head. For example, getting 10 tails in a row followed by a head is a possible, though perhaps unlikely, outcome (TTTTTTTTTH). The sequence of outcomes is \(T^n H\), where \(n\) is any non-negative integer representing the number of tails before the first head.
Forming the Sample Space Set
The sample space, denoted by \(S\), is the set of all these possible outcomes. Listing them out, we get:
\(S = \{H, TH, TTH, TTTH, TTTTH, \dots\}\)
This set includes all sequences that end with the first appearance of H.
Comparing with Options
Let's look at the given options to see which one matches our derived sample space:
- Option 1: {H} - This only includes the case where the first toss is H. It misses all outcomes where one or more tails occur before the first head.
- Option 2: {TH} - This only includes the case where the first toss is T and the second is H. It misses the outcome H and outcomes with more than one tail before H.
- Option 3: {T, HT, HHT, HHHT, ………} - This set seems incorrect. T is not an outcome where the experiment stops with the first head. HT and HHT mean a head occurred before the last head listed, which contradicts the condition of stopping at the *first* head.
- Option 4: {H, TH, TTH, TTTH, ……….} - This set includes H (first toss is head), TH (tail then head), TTH (two tails then head), and so on, indicated by the ellipsis (...). This exactly matches the sample space we derived.
Therefore, the correct sample space for tossing a coin until the first head appears is {H, TH, TTH, TTTH, ...}.
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