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यदि $A , B , C$ किसी यादृच्च्छक प्रयोग के संगत तीन घटनाएँ हों तो सिद्ध कीजिए कि
$P ( A \cup B \cup C )= P ( A )+ P ( B )+ P ( C )- P ( A \cap B )- P ( A \cap C )$
$-P(B \cap C)+P(A \cap B \cap C)$
Solution
Consider $E = B \cup C$ so that
$P ( A \cup B \cup C ) = P ( A \cup E )$
$= P ( A )+ P ( E )- P ( A \cap E )$ …… $(1)$
Now
$P ( E )= P ( B \cup C )$
$= P ( B )+ P ( C )- P ( B \cap C )$ ……… $(2)$
Also $A \cap E=A \cap(B \cup C)$ $=(A \cap B) \cup(A \cap C)$ [using distribution property of intersection of sets over the union]. Thus
$P(A \cap E)=P(A \cap B)+P(A \cap C)$ $-P[(A \cap B) \cap(A \cap C)]$
$= P ( A \cap B )+ P ( A \cap C )- P [ A \cap B \cap C ] $ ……… $(3)$
Using $(2)$ and $( 3 )$ in $(1)$, we get
$P [ A \cup B \cup C ]= P ( A )+ P ( B )$ $+ P ( C )- P ( B \cap C )$ $- P ( A \cap B )- P ( A \cap C )$ $+ P ( A \cap B \cap C )$
