3. Using only the three axioms of probability, prove the Bonferroni inequality: P(AUB P(A) P(B)

Question

Question

Answer 1

Answer #1

Using the third axiom of probability for mutually exclusive events A - B and B, we have

P((A - B) U B) = P(A - B) + P(B)

=> P(A U B) = P(A - B) + P(B)

By second axiom, P(A - B) >= 0

=> P(A U B) >= P(B)

Let us call this inequality 1.

Also we have P(A) + P(A') = P(S).

By first axiom, P(S) = 1

=> P(A) + P(A') = 1

By second axiom, P(A') >= 0

=> 1 >= P(A)

Let us call this inequality 2

Adding inequalities 1 and 2

=> P(A U B) + 1 >= P(A) + P(B)

=> P(A U B) >= P(A) + P(B) - 1.

Answer 2

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