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In probability theory, a sample space is the complete set of all possible outcomes of a random experiment. It's like the foundation of probability, where every possible outcome is listed, ensuring that we account for every eventuality in our analysis. - For instance, if you are tossing a fair coin, the sample space would be \[ \{\text{Heads, Tails}\} \]. This means that when you toss the coin, these are the only outcomes you can expect.- If you're rolling a six-sided die, the sample space is \[ \{1, 2, 3, 4, 5, 6\} \], representing all the numbers that could possibly show up on top.In our exercise, the sample space consists of five simple events: \( E_{1}, E_{2}, E_{3}, E_{4}, \text{ and } E_{5} \). This sample space is crucial because every probability we calculate must be in relation to these five outcomes. Each event denotes a potential individual outcome, all of which collectively represent all that could occur in this scenario.

Simple events are the individual possible outcomes or events that cannot be broken down further. They are the building blocks within a sample space. Each simple event is mutually exclusive, meaning the occurrence of one excludes the others. - For example, in the toss of a coin, "Heads" and "Tails" are simple events. Neither can happen simultaneously, and each is distinct from the other.In the given exercise, the simple events are \( E_{1}, E_{2}, E_{3}, E_{4}, \text{ and } E_{5} \). Each of these is treated as a unique outcome with a probability attached:- The known probabilities given are \( P(E_{1}) = P(E_{2}) = 0.15 \) and \( P(E_{3}) = 0.4 \).- We have the information that \( P(E_{4}) = 2P(E_{5}) \), which shows a relation between these two simple events.Understanding simple events helps us grasp the structure of the sample space, enabling the calculation of probabilities for each potential scenario.

The principle of total probability ensures that the probability of all possible outcomes of an experiment adds up to 1. This is because the sample space includes every potential event, and together they represent certainty—something in that space will occur.- For instance, with a die roll, the probabilities of each outcome \( \{1, 2, 3, 4, 5, 6\} \) must sum to 1.In the given exercise, we're using this principle by summing the probabilities of the simple events in the sample space:- The relationship is illustrated through the equation \[ P(E_{1}) + P(E_{2}) + P(E_{3}) + P(E_{4}) + P(E_{5}) = 1 \].- By substituting and solving, we derived values for \( P(E_{4}) \text{ and } P(E_{5}) \text{ where } P(E_{4}) = 0.2 \text{ and } P(E_{5}) = 0.1 \).Applying total probability not only confirms our calculations but also ensures logical consistency within the sample space. This concept is foundational because it guarantees that every calculation of an event's probability is valid within the context of the entire set of outcomes.