Question

(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent;...

(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that

P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1).

(b) Let A and B1 be independent; similarly, let A and B2 be independent. Show that in this case, A and B1∪B2 are independent if and only if A and B1∩B2 are independent.

(c) Given P(A) = 0.42,P(B) = 0.25, and P(A∩B) = 0.17, find

(i)P(A∪B) ;

(ii)P(A∩Bc) ;

(iii)P(Ac∩Bc) ;

(iv)P(Ac|Bc).

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Complete solution is given in attached images:

а) te have (An U.AZ) C (Ben B27 А, СВ, ОВ, д А. СВ.,ПВ, А, С, А, СВ, А, СВ, А, СВ. А, с А, ОВ, от А = A, B, к А, = A, Пр, отNow A & B Lis independent 3)_P(ANB) = f(A). P (B1) A & B2 is independent = P(ANB) P(A). P (B) if a - Now A & B.B. are indepen- l(A).P (80) + P(A): P[82-PA):P (B,18,)=_ pan P (AN (8,082) 3 pca) [p@.)+ P(6.) - Rebo PCB,18.) = plan (8,08>) = P(A) P (B,e) ii) P (ANB) P(A) - P(ANB) 0.42-0.17 11 0-25 iii) PARB) 1- P(AUB) 1-0.5 0.5 iy) P (A/B) P(AnB? PB9 PA AB) 1- PB 0.50 0.75 0

Thank You.

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(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent;...
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