3个概率公式的证明,Provethefollowingassertions.1.P[A1∩A2∩A3∩.An]>=P[A1]+P[A2]+...+P[An]-(n-1)foranyeventsA1,A2,...An2.-1/4

Question

3个概率公式的证明,Prove the following assertions.
1.P[A1∩A2∩A3∩.An]>=P[A1]+P[A2]+...+P[An]-(n-1) for any events A1,A2,...An
2.-1/4

Answer 1

1.
Using mathematical induction,
when n=1,equality holds;
when n=2,the inequality is just P[A1∪A2]=P[A1]+P[A2]+...+P[A(k-1)]+P[Ak∩A(k+1)]- (k-1)>=P[A1]+P[A2]+...+P[A(k-1)]-(k-1)+P[Ak]+P[A(k+1)]-1=P[A1]+P[A2]+...+P[A(k+1)]-k.
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2.
P[A∩B]-P[A]P[B]