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In probability theory, a sample space is the set of all possible outcomes or sample points of a particular experiment. Each outcome in this set is distinct and represents an individual result of the experiment or process of observation. For example, in the given exercise, the sample space is denoted as \(S = \{E_1, E_2, E_3, E_4, E_5, E_6, E_7\}\). This notation indicates that there are seven distinct events, each of which might occur as a result of the experiment. It is essential to understand that a well-defined sample space is crucial, as it forms the foundation for calculating probabilities of various related events. Each sample point in \(S\) has an assigned probability, such as \(P(E_1) = 0.05\) and \(P(E_2) = 0.20\), indicating the likelihood of occurrence for each specific outcome. Being clear about the sample space ensures that probabilities are assigned correctly and add up to a total of 1.

Mutually exclusive events are events that cannot happen at the same time. They have no sample points in common. If you consider two mutually exclusive events, the occurrence of one event prevents the occurrence of the other. In our given example, events \(A = \{E_1, E_4, E_6\}\) and \(C = \{E_2, E_3, E_5, E_7\}\) are mutually exclusive because they share no elements or outcomes in common. This means that whenever event \(A\) occurs, event \(C\) cannot occur and vice versa.

Understanding this principle helps when calculating combined probabilities, as the probability of their union is simply the sum of their individual probabilities, since they do not overlap. It’s important to identify these types of events early on, as they simplify the analysis and calculation of probabilities in a complex space.

The probability of an event is a measure of the likelihood that the event will occur. It is calculated by summing the probabilities of the individual sample points that make up the event. For example, to find the probability of event \(A\) in the given exercise, we add the probabilities of the sample points in \(A\), which are \(E_1, E_4, \) and \(E_6\). Therefore, \( P(A) = P(E_1) + P(E_4) + P(E_6) = 0.05 + 0.25 + 0.10 = 0.40 \).

Probabilities are always between 0 and 1, where 0 means the event cannot occur and 1 means the event is certain. Calculating probabilities involves adding the probabilities of mutually exclusive events when determining the probability of their union. For instance, the probability of \(A \cup B\) is found by considering all the unique outcomes in both \(A\) and \(B\) and summing their probabilities. By understanding how probabilities of events are determined, one can predict and analyze the likelihood of different scenarios occurring, making probability a powerful tool for decision-making.