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The sample space refers to the set of all possible outcomes of a particular experiment.
In probability theory, it is denoted by the letter \( S \).
For our exercise, the sample space is \( \{E_1, E_2, E_3, E_4, E_5\} \). Each outcome is one of the possible results of the experiment.
To calculate probabilities, it is essential to know the total number of outcomes within the sample space.
When all outcomes are equally likely, as in this case, the probability of each individual outcome is given by \( \frac{1}{n} \),
where \( n \) is the total number of outcomes in the sample space.
This concept helps in determining the probability of any event occurring by analyzing the proportion of the sample space it covers.
In our exercise, since there are five outcomes, each has a probability of \( \frac{1}{5} \).
By understanding the sample space, you can easily deduce the likelihood of any event defined in relation to it.

Mutually exclusive events are events that cannot occur at the same time.
In other words, if one event happens, the other cannot.
In this exercise, events \( A \) and \( B \) are mutually exclusive.
This is because their intersection is the empty set, meaning they have no outcomes in common:

• Event \( A \): \( \{E_1, E_2\} \)
• Event \( B \): \( \{E_3, E_4\} \)

Their intersection \( A \cap B = \emptyset \).
If two events are mutually exclusive, the probability of their union is simply the sum of their individual probabilities:
\( P(A \cup B) = P(A) + P(B) \). This principle reflects the fact that the occurrence of one does not affect the occurrence of the other.
In practical terms, it means when one event's occurrence is confirmed,
there's no need to consider the possibility of the other event occurring simultaneously.
By recognizing mutually exclusive events, you can simplify probability calculations significantly.

The complement of an event refers to all possible outcomes that are not part of that event.
If you have an event \( A \), its complement, \( A^c \), is the set of all outcomes in the sample space that are not in \( A \).
For example, in our exercise:

• Event \( A = \{E_1, E_2\} \)
• Complement \( A^c = \{E_3, E_4, E_5\} \)

This means any outcome not in \( A \) is in \( A^c \).
The probability of the complement can be easily calculated using the formula:
\( P(A^c) = 1 - P(A) \).
This highlights that the total probability of all possible outcomes (events plus their complements) always equals 1.
Understanding event complements is crucial for tackling probability questions that ask for the likelihood of an event not occurring.
Inverted situations often require focusing on the complement rather than the original event itself.

The union of two events includes all the outcomes that are part of either event, or both.
It is represented by the symbol \( \cup \).
To find the union, you combine all unique outcomes from the given events.
In the exercise, consider the union of events \( A \) and \( B \):

• Event \( A = \{E_1, E_2\} \)
• Event \( B = \{E_3, E_4\} \)
• \( A \cup B = \{E_1, E_2, E_3, E_4\} \)

The probability of the union is calculated by considering all outcomes in \( A \cup B \).
If events are mutually exclusive, as in the case of \( A \) and \( B \), this probability is simply the sum of the probabilities of each event:
\( P(A \cup B) = P(A) + P(B) \).
Understanding unions is essential to evaluate scenarios where the occurrence of one or more different events is of interest.
By identifying the union of events, you provide an overall picture of "either-or" occurrences, which can simplify the decision-making process in probabilistic evaluations.
Unions are powerful when combined with other set operations, allowing for complex probability assessments.