Applying Induction to Inclusion-Exclusion Principle for Probability Measures

[Julio1]

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]

I presume that the "$$A_i$$" are sets but what is "P"?

 

[Julio1]

I presume that the "$$A_i$$" 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.:

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