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In probability theory, the **sample space** is the foundation for understanding experiments and events. It represents the set of all possible outcomes of a random experiment. Imagine rolling a die; the sample space would include \{1, 2, 3, 4, 5, 6\}, because these are the possible results of a single roll.

For our exercise, the sample space consists of the outcomes \{a, b, c, d, e\}. Each outcome has a specific probability. Assigning these probabilities ensures that the total probability of the sample space equals 1. Therefore, \[P(a) + P(b) + P(c) + P(d) + P(e) = 1.0\]

The probabilities given are:

• \[P(a) = 0.1\]
• \[P(b) = 0.1\]
• \[P(c) = 0.2\]
• \[P(d) = 0.4\]
• \[P(e) = 0.2\]

Understanding this setup is crucial for calculating any further probabilities of specific events within this sample space.

**Event probability** refers to the likelihood of a specific occurrence within the sample space. An event can be a single outcome or a group of outcomes. To calculate the probability of an event, you sum up the probabilities of all the outcomes that make up the event. In this exercise, we have event \(A = \{a, b, c\}\) and event \(B = \{c, d, e\}\).

**Probability of Event A**: To find \(P(A)\), sum the probabilities of \(a\), \(b\), and \(c\): \[P(A) = P(a) + P(b) + P(c) = 0.1 + 0.1 + 0.2 = 0.4\]

**Probability of Event B**: Similarly, for \(P(B)\), sum the probabilities of \(c\), \(d\), and \(e\): \[P(B) = P(c) + P(d) + P(e) = 0.2 + 0.4 + 0.2 = 0.8\]

Each calculation uses the defined probabilities of the respective outcomes in the sample space.

A **complementary event** in probability refers to all possible outcomes in the sample space that are not part of the original event. It is usually denoted by a prime symbol, such as \(A'\), the complement of event \(A\). The probability of a complementary event is determined by subtracting the event's probability from 1.

In our exercise, event \(A\) consists of outcomes \(\{a, b, c\}\). So, the complementary event \(A'\) would include the remaining outcomes \(\{d, e\}\).

To find \(P(A')\): \[P(A') = 1 - P(A) = 1 - 0.4 = 0.6\]

You could also directly sum the probabilities of outcomes in \(A'\): \[P(A') = P(d) + P(e) = 0.4 + 0.2 = 0.6\]

This approach confirms the correct calculation of complementary event probabilities.

Understanding **union and intersection** helps in calculating probabilities for events with overlapping outcomes. The union of events, denoted as \(A \cup B\), includes all outcomes found in either event A or event B (or both). The intersection of events, \(A \cap B\), focuses only on outcomes common to both events.

**Union of Events**: For the union \(A \cup B\), you sum the probabilities of all distinct outcomes from both events:\[P(A \cup B) = P(a) + P(b) + P(c) + P(d) + P(e) = 1.0\]
Since this covers the entire sample space, \(P(A \cup B)\) equals 1.

**Intersection of Events**: The intersection \(A \cap B\) includes only the common outcome in both \(A\) and \(B\), which is \(c\). Therefore,\[P(A \cap B) = P(c) = 0.2\]

These concepts simplify complex probability scenarios by breaking down events into their shared and combined outcomes. This allows for strategic probability calculus.