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
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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:
\[ S = {1, 2, 3, 4, 5, 6} \]
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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:
\[ E = {2, 4, 6} \]
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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:
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Sample Space: \[ S = {1, 2, 3, 4, 5, 6} \]
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Probability Function: \[ P(x) = \begin{cases} \frac{1}{6} & \text{if } x \in S \ 0 & \text{if } x \notin S \end{cases} \]
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Example Event: Let \(E = {2, 4, 6}\) (rolling an even number).
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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.