Union Bound




     

Following is my proof. I didn’t get full marks with this proof.

At  the end of this post, there is another proof, which gets full
marks.

base case: P(A1) <= P(A1)

P(A1 ∪ A2) = P(A1) + P(A2) ‐ P(A1 ∩ A2) <= P(A1) + P(A2)

P((A1 ∪ A2) ∪ A3) = P(A1 ∪ A2) + P(A3) ‐ P((A1 ∪ A2)  ∩  A3)   <=
P(A1 ∪ A2) + P(A3) <= P(A1) + P(A2) + P(A3)











proof:



Bonferroni Inequality:



proof:



use this: