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Need help with this ASAP10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using the res

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Answer #1

3)

a)

P(A_1^c \cap A_2^c \cap A_3^c) = P[(A_1 \cup A_2 \cup A_3)^c] (By De-Moivres theorem)

= 1 - P(A_1 \cup A_2 \cup A_3)

= 1 - [P(A_1)+P(A_2)+P(A_3)-P(A_1 \cap A_2)-P(A_2 \cap A_3)-P(A_1 \cap A_3)+P(A_1 \cap A_2 \cap A_3)]

= 1 - P(A_1)-P(A_2)-P(A_3)+P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3)-P(A_1 \cap A_2 \cap A_3)

We know that,

P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3) \ge P(A_1 \cap A_2 \cap A_3) (By Venn diagram)

P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3) - P(A_1 \cap A_2 \cap A_3) \ge 0

Hence,

1 - P(A_1)-P(A_2)-P(A_3)+P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3)-P(A_1 \cap A_2 \cap A_3)

1- P(A1) - P(A2) - P(A3)

Hence,

P(A_1^c \cap A_2^c \cap A_3^c) \ge 1 - P(A_1)-P(A_2)-P(A_3)

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