MHB Applying Induction to Inclusion-Exclusion Principle for Probability Measures

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The discussion focuses on applying induction to the Inclusion-Exclusion Principle for probability measures, specifically proving the formula for the probability of the intersection of events. Participants clarify that the "A_i" represent events and "P" denotes the probability measure. The hint suggests using induction on the number of events, n, to establish the relationship. The conversation emphasizes understanding the foundational elements of probability and set theory in this context. The thread ultimately aims to derive the stated probability formula through rigorous mathematical reasoning.
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$P(\displaystyle\bigcap_{i=1}^n A_i)=\displaystyle\sum_{i=1}^n P(A_i)-\displaystyle\sum_{i<j} P(A_i\cup A_j)+\displaystyle\sum_{i<j<k} P(A_i\cup A_j\cup A_k)-\cdots - (-1)^n P(A_1\cup A_2\cup ... \cup A_n).$

Hello, the Hint is use induction on $n$.
 
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I presume that the "A_i" are sets but what is "P"?
 
HallsofIvy said:
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.:

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