Sample spaces are really important in probability and statistics. They help us understand all the possible results of an experiment.

1. What is a Sample Space?

   • A sample space, shown as $S$, includes all the possible outcomes. For example, if you flip a coin, the sample space is $S = \{H, T\}$, which means it can land on heads (H) or tails (T).

2. Probability Assignments:

   • Each result in a sample space can have a specific chance, called probability. The total of all these probabilities is always 1. This means that if you add up the chances of getting each outcome, you get 100%: $\sum_{e \in S} P(e) = 1$

3. Event Relationships:

   • Sample spaces help us organize different types of events, like simple ones and more complicated ones. They also make it easier to figure out probabilities using different methods in statistics.

4. Building Blocks for Inferential Statistics:

   • When we understand sample spaces, we can learn important ideas like random variables and distributions. These ideas are really helpful when we want to test our guesses or hypotheses about data.