Question

A mathematical representation that describes a probability experiment, including the sample space and the probability of each outcome.

Answer 1

In probability theory, a mathematical representation of a probability experiment typically consists of three key components: the sample space, events, and the probability function. Let's define each of these components using a simple example, along with an explanation.

### Example: Rolling a Six-Sided Die

1. **Sample Space (S)**:
The sample space is the set of all possible outcomes of a probability experiment. For a single roll of a six-sided die, the sample space is:

\[
S = \{1, 2, 3, 4, 5, 6\}
\]

2. **Events**:
An event is a subset of the sample space. It can consist of one or more outcomes. For example, an event could be rolling an even number:

\[
E = \{2, 4, 6\}
\]

3. **Probability Function (P)**:
The probability function assigns a probability to each outcome in the sample space. In the case of a fair six-sided die, the probability of each outcome is equal. Therefore, the probability of rolling any specific number (e.g., 1, 2, 3, 4, 5, or 6) is:

\[
P(x) = \frac{1}{6} \quad \text{for } x \in S
\]

For the event \(E\) (rolling an even number), the probability would be the sum of the probabilities of the individual outcomes in the event:

\[
P(E) = P(2) + P(4) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
\]

### Summary Representation

Putting this all together, you can represent the probability experiment of rolling a fair six-sided die as follows:

- **Sample Space**:
\[
S = \{1, 2, 3, 4, 5, 6\}
\]

- **Probability Function**:
\[
P(x) =
\begin{cases}
\frac{1}{6} & \text{if } x \in S \\
0 & \text{if } x \notin S
\end{cases}
\]

- **Example Event**:
Let \(E = \{2, 4, 6\}\) (rolling an even number).

- **Probability of Event E**:
\[
P(E) = \frac{1}{2}
\]

This structure serves as a foundational framework for analyzing probability experiments in a systematic and mathematical way.