Independence Extended. Three events A, B, and C are said to...

Independence Extended. Three events $A, B,$ and $C$ are said to be independent if $$ \begin{aligned} P(A \& B) &=P(A) \cdot P(B) \\ P(A \& C) &=P(A) \cdot P(C) \\ P(B \& C) &=P(B) \cdot P(C), \text { and } \\ P(A \& B \& C) &=P(A) \cdot P(B) \cdot P(C) \end{aligned} $$ What is required for four events to be independent? Explain your definition in words.

Independence Extended. Three events $A, B,$ and $C$ are said to be independent if
$$
\begin{aligned}
P(A \& B) &=P(A) \cdot P(B) \\
P(A \& C) &=P(A) \cdot P(C) \\
P(B \& C) &=P(B) \cdot P(C), \text { and } \\
P(A \& B \& C) &=P(A) \cdot P(B) \cdot P(C)
\end{aligned}
$$
What is required for four events to be independent? Explain your definition in words.

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Key Concepts

Mutual Independence

Mutual independence in probability means that every possible subset of a collection of events must satisfy the condition of independence. For four events, this means not only must each pair of events be independent, but every triplet and the complete set of all four events must also have their joint probability equal to the product of their individual probabilities.

Subset Independence

The concept of subset independence requires that for any group of events taken from the four, the probability of their simultaneous occurrence equals the product of their probabilities. This ensures that no combination of events provides additional information about the occurrence of the others.

Product Rule for Independent Events

The product rule is the central principle in defining independent events, stating that if events are independent, then the probability of their intersection is the product of their individual probabilities. Extending this rule to four events means that every possible intersection—from pairs to triples to all four—must satisfy this condition.

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