The correct sample space when flipping two coins is { HH, HT, TH, TT }.
Understanding Sample Space
In the realm of probability, the sample space of an event is defined as the complete set of all possible outcomes that can occur when an experiment is performed. For a simple event like flipping a coin, there are two distinct outcomes:
- H: Represents a Head
- T: Represents a Tail
Deriving the Sample Space for Two Coins
When you flip two coins, the outcome of each coin is independent of the other. To construct the full sample space, we consider every possible combination resulting from the two flips.
Let's break down the possible outcomes for each coin and their combined results:
- First Coin (Coin 1): Can be Head (H) or Tail (T)
- Second Coin (Coin 2): Can be Head (H) or Tail (T)
Combining these possibilities yields the following four unique outcomes:
- HH: The first coin lands on Heads, and the second coin also lands on Heads.
- HT: The first coin lands on Heads, and the second coin lands on Tails.
- TH: The first coin lands on Tails, and the second coin lands on Heads.
- TT: The first coin lands on Tails, and the second coin also lands on Tails.
Therefore, the comprehensive sample space (S) for flipping two coins is explicitly stated as:
S = { HH, HT, TH, TT }
This can be clearly illustrated in the following table:
| Coin 1 Outcome | Coin 2 Outcome | Combined Outcome |
|---|---|---|
| Head (H) | Head (H) | HH |
| Head (H) | Tail (T) | HT |
| Tail (T) | Head (H) | TH |
| Tail (T) | Tail (T) | TT |
Why is This Important?
Establishing the sample space is a fundamental step in calculating the probabilities of various events. With a clear list of all possible outcomes, you can precisely determine the likelihood of specific results. For example:
- Probability of getting exactly one head: This event includes outcomes {HT, TH}. There are 2 favorable outcomes out of 4 total, so the probability is 2/4, or 1/2.
- Probability of getting at least one head: This includes outcomes {HH, HT, TH}. There are 3 favorable outcomes out of 4 total, so the probability is 3/4.
- Probability of getting no heads (both tails): This event is represented by {TT}. There is 1 favorable outcome out of 4 total, so the probability is 1/4.
The sample space provides a complete and systematic foundation for understanding and analyzing probabilistic events.
