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In probability theory, a probability space is a foundational concept.It forms the framework for measuring uncertainties in various scenarios.A probability space consists of three main components: a sample space, events, and a probability measure.
Start by understanding the sample space, which includes all possible outcomes of a random experiment. For example, in the ORIGINAL EXERCISE, the sample space is given as \( S = \{o_{1}, o_{2}, o_{3}, o_{4}\} \).The sample space is crucial as it sets the stage for everything else we calculate within probability theory.

• Events: These are subsets of the sample space. They represent outcomes or combinations of outcomes of interest.
• Probability Measure: This is a function that assigns a probability value to each event, ensuring all probabilities add up to one.

Understanding these basics helps as they ground you in how the theory allows us to predict and manage randomness effectively.

The concept of a sample space is central to probability theory.A sample space is the set of all possible outcomes in a probability experiment.For instance, take a coin flip where the result can be Heads or Tails—the sample space is \( \{H, T\} \).
The sample space lays out all possibilities clearly and systematically.In the given exercise, the sample space \( S = \{o_{1}, o_{2}, o_{3}, o_{4}\} \) outlines four distinct, potential outcomes.

• A finite sample space like this one entails a limited number of outcomes. Each one is distinct and disjoint.
• Understanding how outcomes are structured within the sample space is pivotal to assigning probabilities correctly.

The sample space is not only a set but a guide that dictates how events are formed and probabilities are distributed across those events.

A probability assignment distributes likelihood values across a sample space such that each outcome receives a specific probability.In our example, probabilities are assigned to each outcome \( o_{1}, o_{2}, o_{3}, \text{and} \ o_{4} \) within the sample space.
The allocated probabilities must fulfill two essential conditions:

• Each probability is a number between 0 and 1.
• The sum of all probabilities in a sample space equals 1.

Consider part (a) of the exercise: Given \( \operatorname{Pr}(o_{1}) = 0.15 \) and the condition that \( \operatorname{Pr}(o_{1}) + \operatorname{Pr}(o_{2}) = \operatorname{Pr}(o_{3}) + \operatorname{Pr}(o_{4}) \), solving this requires a logical step-by-step approach.
Similarly, in part (b), knowing that \( \operatorname{Pr}(o_{3}) = 0.22 \) directs how the remaining probabilities are dispersed.This step involves mathematical reasoning to ensure every condition of the probability space is accurately met.