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Conditional Probability helps us understand how likely an event is, given that we already know another event has occurred. It is expressed as \( P(A \mid B) \), which reads as the probability of event A happening given that B has occurred. This concept can clarify relationships between phenomena by considering certain conditions or restrictions.

To calculate conditional probability, use the formula:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)

Here, \( P(A \cap B) \) is the joint probability of both A and B happening, while \( P(B) \) represents the probability of event B.

In our exercise, we used the known conditional probability \( P(Y_2 \mid X_1) \) to find the joint probability. This connection forms the base for understanding many associative relationships in probability theory, helping in tasks like risk assessment or predictive modeling.

Probability rules provide a framework for calculating and understanding how different probability events relate to each other. These rules ensure that the probability of any event is always between 0 and 1. When working with multiple events, certain rules guide us.

Some common probability rules include:

• Addition Rule: If two events, A and B, cannot happen at the same time (disjoint), their combined probability is \( P(A \cup B) = P(A) + P(B) \).
• Multiplication Rule: Applied when determining joint probability, such as identifying probabilities that occur simultaneously or are dependent on each other.
• Complement Rule: The probability of an event not occurring is \( 1 - P(A) \).

These essential probability rules are crucial in solving complex problems by determining how different events interact and impact one another. They also set the foundation for conditional probability.

The Multiplication Rule is a cornerstone in probability calculus, essential for calculating joint probabilities. This rule is particularly useful when dealing with situations of dependent or sequential events.

To find the joint probability \( P(A \cap B) \) of two events, A and B, occurring together, use:

• \( P(A \cap B) = P(A) \times P(B \mid A) \)

Here, \( P(B \mid A) \) indicates that B happens given that A has already occurred. This formula aligns perfectly when events are connected sequentially or one event influences the likelihood of another.

In the given exercise, we used the Multiplication Rule to determine that \( P(X_1 \cap Y_2) = P(Y_2 \mid X_1) \times P(X_1) \). By multiplying these probabilities, we obtained the joint probability of the events \( X_1 \) and \( Y_2 \), delivering the final result of the exercise.