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Please type if possible. 1. For two events A and B show that P(A∩B) ≥ P(A)+P(B)−1....

Please type if possible.

1. For two events A and B show that P(A∩B) ≥ P(A)+P(B)−1. (Hint: Apply de Morgan’s law and then the Bonferroni inequality).

2. Derive below Results 1 to 4 from Axioms 1 to 3 given in Section 2.1.2 in the textbook.

Result 1: P(Ac ) = 1 − P(A)

Result 2 : For any two events A and B, P(A∪B) = P(A)+P(B)−P(A∩B)

Result 3: For any two events A and B, P(A) = P(A ∩ B) + P(A ∩ Bc )

Result 4: If B ⊂ A, then A ∩ B = B. Therefore, P(A) − P(B) = P(A ∩ Bc ), and P(A) > P(B).

Math   Statistics-And-Probability   
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