Probability - Conditional Probability and Multiplication Theorem

Conditional Probability Definition: The probability of an event $A$ occurring given that event $B$ has already occurred is denoted by $P(A|B)$. Visually, this is interpreted as a reduction of the sample space from the entire set $S$ to just the set $B$. On a Venn diagram, we calculate the probability by finding the ratio of the area of the intersection $A \\cap B$ to the total area of the set $B$.

The Multiplication Theorem: This theorem provides a formula for the probability of the simultaneous occurrence of two events $A$ and $B$. It states that $P(A \\cap B) = P(A) \\cdot P(B|A)$, meaning the probability of both events occurring is the product of the probability of the first event and the conditional probability of the second event given that the first has occurred.

Independent Events: Two events are independent if the occurrence of one does not change the probability of the other. Visually, in a Venn diagram, independence implies that the proportion of circle $A$ that overlaps with $B$ is the same as the proportion of $A$ in the entire sample space $S$. Mathematically, this is expressed as $P(A|B) = P(A)$ and $P(B|A) = P(B)$.

Condition for Independence: Two events $A$ and $B$ are independent if and only if $P(A \\cap B) = P(A) \\cdot P(B)$. This serves as the primary test for independence in problems. If this equality does not hold, the events are dependent, and the general multiplication theorem must be used.

Multiplication Theorem for Three Events: The theorem extends to multiple events. For three events $A, B,$ and $C$, the probability of their joint occurrence is $P(A \\cap B \\cap C) = P(A) \\cdot P(B|A) \\cdot P(C|A \\cap B)$. This describes a chain of dependencies, which can be visualized as moving through three levels of a tree diagram.

Tree Diagrams: A tree diagram is a visual representation of multi-stage experiments. Each node splits into branches representing outcomes, and each branch is labeled with its conditional probability. To find the probability of a specific outcome sequence (path), you multiply the probabilities along the branches from the root to the leaf.

Complementary Conditional Probability: For any event $A$ and a given condition $B$, the probability of the complement event $A'$ occurring given $B$ is $P(A'|B) = 1 - P(A|B)$. This rule highlights that within the restricted sample space of $B$, the sum of probabilities of all mutually exclusive and exhaustive events must equal 1.