Conditional Probability & Independence

Overview

When an event has already occurred, the probabilities of some random events may change due to the increase in known information. For example, in a mobile game gacha system, we might initially think that the probability of drawing a six-star character in a single pull is equal to not drawing one. However, after pulling 50 times without a single six-star, stubbornly believing that "the probability of drawing a six-star is equal to not drawing one" would not be wise.

In summary, it is necessary to study the probability of events occurring under certain known conditions.

Conditional Probability

Definition

If it is known that event \(A\) has occurred, the probability of event \(B\) occurring under this condition is called the conditional probability, denoted as \(P(B|A)\).

In the probability space \((\Omega, \mathcal{F}, P)\), if event \(A \in \mathcal{F}\) satisfies \(P(A) > 0\), then the conditional probability \(P(\cdot|A)\) is defined as

\[ P(B|A) = \frac{P(AB)}{P(A)} \quad \forall B \in \mathcal{F} \]

It can be verified that \(P(\cdot|A)\) defined by the above formula is a probability function on \((\Omega, \mathcal{F})\).

Based on the definition of conditional probability, the following two equalities can be directly derived:

Probability Multiplication Formula: In the probability space \((\Omega, \mathcal{F}, P)\), if \(P(A) > 0\), then for any event \(B\),

\[ P(AB) = P(A)P(B|A) \]

Law of Total Probability: In the probability space \((\Omega, \mathcal{F}, P)\), if a set of events \(A_1, \cdots, A_n\) are pairwise disjoint and their union is \(\Omega\), then for any event \(B\),

\[ P(B) = \sum_{i=1}^{n} P(A_i)P(B|A_i) \]

Bayes' Formula

Generally, suppose the possible causes for event \(B\) are \(A_1, A_2, \cdots, A_n\). When \(P(A_i)\) and \(P(B|A_i)\) are known, the probability of event \(B\) occurring can be calculated using the law of total probability. However, in many cases, we need to infer the probabilities of the various cause events based on the result that "event \(B\) has occurred." Thus, we have

\[ P(A_i|B) = \frac{P(A_iB)}{P(B)} = \frac{P(A_i)P(B|A_i)}{\sum_{j=1}^{n} P(A_j)P(B|A_j)} \]

The above formula is Bayes' formula.

Independence of Events

In the study of conditional probability, situations may arise where \(P(B|A) = P(B)\). Intuitively, this means that whether event \(B\) occurs does not provide any information about event \(A\), i.e., event \(B\) is "unrelated" to event \(A\). This leads to the following definition.

Definition

If events \(A\) and \(B\) in the same probability space satisfy

\[ P(AB) = P(A)P(B) \]

then \(A\) and \(B\) are said to be independent. For multiple events \(A_1, A_2, \cdots, A_n\), we say they are independent if and only if for any subset of events \(\{ A_{i_k} : 1 \leq i_1 < i_2 < \cdots < i_k \leq n \}\),

\[ P( A_{i_1}A_{i_2} \cdots A_{i_r} ) = \prod_{k=1}^{r} P(A_{i_k}) \]

Independence of Multiple Events

For multiple events, pairwise independence generally does not imply mutual independence. Consider the following counterexample:

There is a regular tetrahedron die, with three faces painted red, green, and blue respectively, and the fourth face containing all three colors. Now, roll the die once, and let events \(A\), \(B\), \(C\) represent the face in contact with the table containing red, green, and blue, respectively.

It is easy to calculate \(P(A) = P(B) = P(C) = \frac{1}{2}\), while \(P(AB) = P(BC) = P(CA) = P(ABC) = \frac{1}{4}\).

Clearly, \(A, B, C\) are pairwise independent, but since \(P(ABC) \neq P(A)P(B)P(C)\), \(A, B, C\) are not mutually independent.

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