Show that for any three events A, B, C with P(C) > 0 P(A ? B|C) = P(A|C) + P(B|C

Question

Show that for any three events A, B, C with P(C) > 0 P(A ? B|C) = P(A|C) + P(B|C) ? P(A ? B|C). --------------- 8. Suppose that events A and B are independent. (a) Show that A and Bc are independent. (b) Show that Ac and Bc are independent.------------- 9. Suppose that P(A|B) = P(A|Bc) (P(B) > 0 and P(Bc) > 0 both understood). Show that A and B are independent.-------------- 10. True or false: For events A,B,C, if (i) A and C are independent and (ii) B and C are independent, then it follows that A ? B and C (as a pair of events) are independent. If true, provide your reasoning. If false, provide an example (pictured as a Venn diagram) in which (i) and (ii) are true, but A ? B and C are not independent.

Explanation / Answer

P(C) >0
P( A U B | C)
= P(( A U B) n C)/ P (C)
= P(( A n C) U (B n C))/ P(C)
=P( A n C)/ P(C)+ P( B n C) / P(C) + P(( A n C) n ( B n C))/ P(C)
= P( A|C)+ P( B| C)- P(A n B n C)/ P(C)
=P( A|C)+ P( B| C)- P(A n B | C)

Hence proved this part.

8) A and B are independent iff P( A n B)= P(A)* P(B)
a)P( A n BC)= P (A)- P( A n B)= P(A)- P( A) P(B)= P(A)*(1-P(B))= P(A)*P(Bc).

hence proved part A

b) P( A C n B C)= P( Bc)- P( A n BC)= P(Bc)- P(A)*P(Bc) [ from part a]

= P(Bc)*(1-P(A))= P(Bc)*P(Ac)

hence proved part b.

9)P( A| B)= P( A| BC)

=> P( A n B)/ P(B)= P(A n Bc)/ P(Bc)= (P(A)- P(A n B))/(1-P(B))

=>P(A n B)- P(A n B)* P(B)= P(A)*P(B)-P(A n B)* P(B)

=> P(A n B)= P(A)*P(B)

Hence proved A and B are independent.

10)

The answer is false.

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