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Combinatorics is the field of mathematics focused on counting, arrangements, and combinations. It helps us calculate the number of different ways things can happen when dealing with discrete structures. In this problem, we use combinatorics to figure out how vehicles can turn in different directions at an intersection.
For example, if each vehicle can make one of three choices—Right (R), Left (L), or Straight (S)—then for three vehicles, we calculate possible outcomes as \(3 \times 3 \times 3 = 27\). This multiplication represents the product rule, where the number of outcomes for each vehicle is multiplied together. Combinatorics becomes essential when solving for various events and exploring all possible outcomes in situation-based problems like these.

An outcome space is the entirety of possible results in a probabilistic event. This term is crucial in probability because it defines the universe from which events are considered. In the freeway exit problem, the outcome space consists of every combination of turns three vehicles can make.
The space includes sequences like "RRR" (all turn right) or "RLS" (each vehicle chooses a different direction). These sequences highlight that each vehicle's choice influences the total outcome space of 27 distinct combinations—each a string of three letters representing their respective turns. Understanding outcome spaces gives us the groundwork for further decomposition into events and more intricate probability calculations.

Events are specific sets of outcomes that we are interested in. When we talk about sample spaces, we mean the complete set of all possible outcomes.

• Event A (All Same Direction): This event is concerned with outcomes like "RRR," "LLL," and "SSS," where all three vehicles turn in the same direction.
• Event B (Different Directions): Here, we have outcomes such as "RLS," "RSL," where each vehicle chooses a distinct direction.
• Event C (Exactly Two Right Turns): In this case, outcomes are "RRL," "RRS," showing two right turns.
• Event D (Exactly Two Same Direction): This keeps track of outcomes like "RRL," "RRS," where exactly two directions match.

Understanding how to identify and count these events relative to the sample space is critical, especially when refining and analyzing the probability of more complex questions in the exercises.

Set operations help us analyze and simplify probability problems by combining different events. Here, we use basic set operations like union, intersection, and complement to evaluate complex scenarios.

• Union (\(C \cup D\)): This operation combines outcomes from Events C and D, presenting all outcomes that belong to either one or both events. Here, it results in unique outcomes like "RRL," "LLR," "SRR."
• Intersection (\(C \cap D\)): The intersection finds common outcomes between Events C and D. For this problem, include outcomes such as "RRL," "RRS," where the scenarios overlap exactly.
• Complement (\(D'\)): This is the set of outcomes not included in Event D, showing sequences where not exactly two turn the same way.

These operations are vital because they allow us to build more complex probability models by understanding how different events interact and overlap, providing a basis for complex decision-making processes.