Question

Paragraph Styles (a) Use a Venn diagram in which the event areas are proportional to their probabilities to illustrate three non-trivial events A, B, and C that are independent. (b) Provide, if possible, another solution to part (a) that also has P[A] = 0.4 and P[AC] = 0.1. (c) Using a probability table, illustrate three non-trivial events A, B, and C that are pairwise independent but not overall independent (i.e., P[ABC] 6= P[A]P[B]P[C]).

Answer 1

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Solution: a) Let Given A, 8c are indepescent. PCA) = . 4 p (0. 5 PCC): 3 Plano) - PIA) PIS. = .4 x .5 = "20 planc) - PINX.pcc). :-•4 *•3 - 12, PC BOC) = PIC3X PCC) 3.5% •3 =15 PLADBOC) - PLA)XPLB) x ⓇCC) sol *.5%*3 = 806 D

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b) Give P(A)=-4 planc) = 1 gives oney scepers derst Picane) - PCA) *PCC). Hoad gives :12:44 * X pec) = 125 Let (13) = -5. PLANB) = M(A) * PIC) = 47.5.202 plane) = . Plon) - 15x•25 = 12 PCAMBOC) = P(ADXPLB), PCC) = .4%. 5x:25 2:05

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AC C=%3 The is 3x3, with dait now em sample space => ASC =0 - In the venn chicay neang . assume ole Snowc hay Na Corporate Probabil .. ieg PLA=% 3 Area CAI=/ PCBD-49 , Areacos y pe=% g Area Co2 = % The three way 1 Intersection Intersection Agc no ARC has zeno Saobabiun ieg PCABC) = 0 + = As, and dent. PLA)xpeB) & peco are not munan my indepen- S H Ping) = plod - peac) = 19. i e, PLAB) = ²*1/ / P(BC) = P(B) x 8 PEAC) = pe A)*P(0) = 49 - Ag and c are pairwise independent.