(Solution) - As in Section 6 3 Exercise 5 and Section 6 4 Exercise -(2025 Original AI-Free Solution)

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As in Section 6.3, Exercise 5 and Section 6.4, Exercise 5, the sample space is S = {0, 1, 2, 3, 4}, Pr({0}) = 0.2, Pr({1}) = 0.3, Pr({2}) = 0.4, Pr({3}) = 0.1, Pr({4}) = 0.0, A = {0, 1, 2}, and B = {0, 2, 4).
Check whether the above event is independent by checking three equations:
Pr(A) = Pr(A | B) A is independent of B
Pr(B) = Pr(B | A) B is independent of A
Pr(A n B) = Pr(A)Pr(B). The multiplication rule
Do you ever find a case where only one or two of these equations is satisfied?