Union and Intersection of events, Mutually Exclusive

Union and Intersection

Definition 1. For any events $E, F$, we define the union of $E$ and $F$ by $E \cup F$. Similarly, the intersection of $E$ and $F$ is defined as $EF = E \cap F$. 

두 사건 $E, F$가 있을 때 이 사건 둘 중 하나가 혹은 둘 다 일어나는 사건을 union, 합사건이라고 하고, 둘 다 일어나는 사건을 intersection, 곱사건이라고 부른다. 숫자를 늘려서 다음과 같이 무한합, 무한곱도 정의할 수 있다.

 

Definition 2. Let $E_1, E_2, ...$ be events. We define the union of these events, denoted by $\cup_{n=1}^{\infty} E_n$, by the event that consists of all outcomes that are in $E_n$ for at least one value of $n = 1, 2, ...$. Similarly, we define the intersection of the events $E_n$, denoted by $\cap_{n=1}^{\infty} E_n$, by the event consisting of those outcomes that are in all of the events $E_n, n= 1, 2, ...$.

 

Mutually Exclusive

Definition 3. If any event $E$ is an empty set, we call $E$ the null event and denote it by $\emptyset$. For any events $E, F$, if $EF = \emptyset$, then $E$ and $F$ are said to be mutually exclusive. 

두 사건의 곱사건이 null event, 즉 영사건이라면 두 사건은 결코 동시에 일어나지 않으므로 mutually exclusive, 상호 배반이라고 한다. 

 

Proposition

Proposition. For an event $E, F$,
(1) $P(E) + P(E^c) = 1$, or $P(E^c) = 1 - P(E)$
(2) If $E \subset F$, then $P(E) \leq P(F)$.
$(\because)$ Note that $F = E \cup E^cF$. Because $E$ and $E^cF$ are mutually exclusive, from Axiom (3), we obtain $P(F) = P(E) + P(E^cF)$ which proves the result.
(3) $P(E \cup F) = P(E) + P(F) - P(EF)$
$(\because)$ Note that $E \cup F = E \cup FE^c$. Then we have $P(E \cup F) = P(E) + P(FE^c)$. Because $F = EF \cup FE^c$, $P(E \cup F) = P(E) + P(F) - P(EF)$.

Proposition (3)은 the inclusion-exclusion identity, 포함 배제 원리라고도 알려져 있으며, 다음과 같이 일반화 할 수 있다.

 

The Inclusion-Exclusion Identity

For any sequence of event $E_1, ..., E_n$, $$P(\bigcup_{i=1}^n P_i) = \sum_{i=1}^n P(E_i) - \sum_{i_1 < i_2} P(E_{i_1}E_{i_2}) \\ + \cdots + (-1)^{r+1} \sum_{i_1 < \cdots < i_r} P(E_{i_1} \cdots E_{i_r}) \\ + \cdots + (-1)^{n+1} P(E_1 \cdots E_n)$$