Defining Theoretical Probability and Sample Space

Property A sample space is the list of all possible outcomes for a probability experiment. An event is a subset of the sample space. Theoretical probability is calculated as: $$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$$ A uniform probability model assigns equal probability to all outcomes in the sample space.

Examples When rolling a standard six sided die, the sample space is $\{1, 2, 3, 4, 5, 6\}$. The probability of the event 'rolling an even number' is $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$. When flipping a coin, the sample space is $\{\text{heads}, \text{tails}\}$. This is a uniform model where $P(\text{heads}) = P(\text{tails}) = \frac{1}{2}$. In a class of 20 students, if a student is selected at random, the probability that any specific student like Sam is selected is $P(\text{Sam}) = \frac{1}{20}$.

Explanation Understanding sample spaces and theoretical probability provides the foundation for analyzing more complex situations. By identifying all possible outcomes and counting favorable ones, we can calculate exact probabilities for events. This theoretical approach will be essential when working with compound events involving multiple steps or conditions.