91Ó°ÊÓ

Set theory is a fundamental part of understanding probability and mathematics as a whole. It deals with the collection of objects, known as elements, which can be grouped together into sets. In probability, these elements usually represent possible outcomes of an experiment or event.

When we talk about sets, we often represent them using curly braces, like \(S = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\}\). Each pair in this set \(S\) is a unique outcome combining Al's and Bill's choices for the election. For example, \( (1,2) \) represents the outcome where Al chooses candidate 1 and Bill chooses candidate 2.

• Each outcome is a distinct element in the set \(S\).
• These pairings are exhaustive, covering every possible scenario of Al and Bill choosing from candidates 1, 2, and 3.

Set theory helps us keep track of these possible outcomes, which is crucial for analyzing more specific events or conditions, such as both voters voting the same way.

Combinatorics involves counting, arranging, and combining objects in a systematic way, and it's an essential tool in probability theory. In this exercise, we calculate the number of possible combinations of candidates that Al and Bill can choose. Since each has 3 candidate options, we multiply these choices: \(3 \times 3\) to find the total possible outcomes.

This multiplication gives us 9 unique outcomes, creating the entire sample space:

• Choices: Three candidates (1, 2, or 3)
• Total combinations: \(3 \times 3 = 9\) possibilities

Combinatorics allows us to systematically list and explore complex scenarios involving multiple decision points, such as which candidates Al and Bill can each vote for. This understanding makes it easier to break down and analyze specific events and their probabilities.

Statistical events describe specific outcomes or sets of outcomes within a larger sample space. In this exercise, we are interested in two events, **Event A** and **Event B**.

**Event A** occurs when Al and Bill choose the same candidate. This narrows down the outcomes to \( (1,1), (2,2), (3,3) \), where each pair represents them both selecting the same candidate. Such a shared choice can have implications in analyzing voter behavior assumptions or predictions.

**Event B** involves Al and Bill both choosing candidates other than candidate 2. Thus, the pairs of outcomes meeting this condition are \( (1,1), (1,3), (3,1), (3,3) \). This kind of event is useful for studying the impact of other candidates being selected and is crucial for understanding biases in choice.

These events are central to probability as they help define what outcomes we are interested in and aid in making predictions about real-world behaviors.