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In probability, the concept of a sample space is like having a complete map of all possible outcomes for a given experiment. It's the foundation that allows us to calculate probabilities and make sense of what might happen when we conduct the experiment. The sample space is denoted by the symbol \( S \) and includes every possible event that can occur. For example, if you are rolling a six-sided die, the sample space is \( S = \{ 1, 2, 3, 4, 5, 6 \} \). Every possible number that the die can land on represents a "simple event." Understanding sample spaces is crucial because it ensures that we account for all possibilities, which is necessary when calculating the probabilities of specific events happening. Without knowing the sample space, it's like trying to solve a puzzle without all the pieces.

A simple event is a concept in probability that represents a single outcome within a sample space. Each simple event is distinct and doesn't depend on any other event's outcome. In our problem, the simple events are \( E_1, E_2, E_3, E_4, \) and \( E_5 \), each with its own probability indicating how likely it is to occur. Simple events are important building blocks in probability because they allow us to understand and predict the likelihood of more complex combinations of events. For instance, when we talk about event \( A \) consisting of \( E_1, E_3, \) and \( E_4 \), it is actually formed by combining these simple events. This helps in constructing probabilities for more compound events by systematically summing their components.

Calculating the probability of an event involves summing the probabilities of all simple events that comprise it. In probability theory, the likelihood of different events is expressed as a number between 0 and 1. Numbers closer to 1 indicate that an event is more likely to happen, whereas numbers near 0 suggest it is unlikely. For instance, to find the probability of event \( A \), which includes simple events \( E_1, E_3, \) and \( E_4 \), we add their respective probabilities: \[ P(A) = P(E_1) + P(E_3) + P(E_4) = 0.15 + 0.4 + 0.3 = 0.85 \] Similarly, the probability of event \( B \) consisting of \( E_2 \) and \( E_3 \) is calculated as: \[ P(B) = P(E_2) + P(E_3) = 0.15 + 0.4 = 0.55 \] These calculations allow us to determine the likelihood of more complex events occurring, making probability a highly predictive and useful tool.

Set operations are fundamental in probability as they help us define and calculate probabilities of combined events. In particular, union and intersection operations are crucial when dealing with probabilities. The **union** of two events, labeled as \( A \cup B \), represents all simple events that are in either \( A \) or \( B \) or both. In the exercise, the simple events from the union of \( A \) and \( B \) are \( E_1, E_2, E_3, \) and \( E_4 \). The **intersection** of two events, denoted by \( A \cap B \), contains only those simple events that are in both \( A \) and \( B \). Here, the intersection is simply \( E_3 \) as it is the only event common to both. These set operations simplify the calculation and understanding of probabilities for complex events, making them an indispensable part of probability theory.