Question

a. Prove: If A and B are independent, then so are A and B.

b. Prove: If A and B are independent, then so are $\bar{A}$ and $\bar{B}$.

c. Give an example of events A, B, and C such that

but

P(An Bn C) = P(A)P(B)P(C), P(AnBn C) P(A)P(B)P(C)

Answer 1

Poga NO O Solution rdaperdunt A and B ae a, PCAJ. PLB) PCA AB) A deperdtert c P(A) PCANB PAAB) PCA) P(A) PB) pCA) - B PIA). PCG) p(AnB) indperdunt events e ALE everd tuo indeperdent A Suffose idipedent evert aso p(A PLB PCANB) = PLAUB P(AAE) (aw P(AUB) P (n) CPCA)PB)P(A8) pla) - pl6) P (An8) p(B)-P(A 1- P(A) + /- PA) C-pB] PCA PCE) P CA AB) evert inde perdert

Paga NO -(2 rdapirdant events ot A,B ard c C tfat Such PCANBAC) PCA) PCB) PCC) 2 tau Cay everdt us Conside Cot an Saunple Aawing Outame event of auna ut A be Coiny tRo 2 A - PCA) haod in event agetg to st on teo be pCB) heod eert af gethrg to t Con c be Po ut CTH) (T 2. Than p(AnB) pla) PCB) PCB) pcc) P(BnC) P(A) PCC) t PCAAC) PLA) pCB) PIC). But P(AnBnc)