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Inclusion-exclusion principle

Julio

Member
Feb 14, 2014
71
Show that

$P(\displaystyle\bigcap_{i=1}^n A_i)=\displaystyle\sum_{i=1}^n P(A_i)-\displaystyle\sum_{i<j} P(A_i\cup A_j)+\displaystyle\sum_{i<j<k} P(A_i\cup A_j\cup A_k)-\cdots - (-1)^n P(A_1\cup A_2\cup ... \cup A_n).$

Hello, the Hint is use induction on $n$.
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
I presume that the "[tex]A_i[/tex]" are sets but what is "P"?
 

Julio

Member
Feb 14, 2014
71
I presume that the "[tex]A_i[/tex]" are sets but what is "P"?
Hello HallsofIvy. It is understood that $A_i$ are events and $P$ is a measure of probability, i.e.:

$P: \mathcal{A}\to [0,1], A\mapsto P(A).$