Probability - Independent Events

Definition of Independence: Two events $A$ and $B$ are said to be independent if the occurrence or non-occurrence of one does not affect the probability of the occurrence of the other. Visually, in a Venn diagram, independence is not shown by disjoint circles but by a specific proportional relationship where the area of the intersection $P(A \cap B)$ is exactly the product of the areas of $A$ and $B$.

The Multiplication Theorem: For independent events, the probability of both events occurring simultaneously is the product of their individual probabilities, expressed as $P(A \cap B) = P(A) \times P(B)$. In a tree diagram, this is visualized by multiplying the probabilities along the specific branches that lead to the desired outcome.

Conditional Probability Property: If $A$ and $B$ are independent, the conditional probability of $A$ given $B$ is simply the probability of $A$, i.e., $P(A|B) = P(A)$. This implies that the 'reduced sample space' $B$ contains $A$ in the same proportion as the original sample space $S$ contains $A$.

Independence of Complements: If events $A$ and $B$ are independent, then their complements are also independent. This means pairs like $(A' \text{ and } B)$, $(A \text{ and } B')$, and $(A' \text{ and } B')$ also satisfy the multiplication rule. This is often visualized in a contingency table where row and column totals maintain consistent ratios.

Mutual Independence of Multiple Events: Three events $A, B,$ and $C$ are mutually independent if they are pairwise independent (e.g., $P(A \cap B) = P(A)P(B)$) AND the probability of all three occurring is the product of all three probabilities: $P(A \cap B \cap C) = P(A)P(B)P(C)$. This ensures that no combination of events provides information about another.

Sampling with Replacement: A practical visual scenario for independent events is 'Sampling with Replacement'. When a ball is drawn from an urn and put back before the next draw, the composition of the urn remains unchanged. In a probability tree, this results in identical probability values on branches at every level of the experiment.

Distinction from Mutually Exclusive Events: It is vital to note that independent events are different from mutually exclusive events. Mutually exclusive events cannot happen at the same time ($P(A \cap B) = 0$), whereas independent events usually have a non-zero intersection area. Visually, mutually exclusive events are separate circles, while independent events overlap.