Two events A and B are such that 𝑃(𝐴) = 0.45, 𝑃(𝐵) = 0.22 𝑎𝑛𝑑 𝑃(𝐴 ∪ 𝐵) = 0.53.
5.1.1 Find 𝑃(𝐴̅∪ 𝐵̅)
5.1.2 Are the events A and B independent?
We can present the union of events as: "A\\cup B=(A\\backslash(A\\cap B))\\cup(A\\cap B)\\cup(B\\backslash(A\\cap B))". Thus, we get: "P(A\\cup B)=P(A\\backslash(A\\cap B))+P((A\\cap B))+P(B\\backslash(A\\cap B))=P(A)+P(B\\backslash(A\\cap B))".
From the latter we find: "P(A\\cup B)-P(A)=P(B\\backslash(A\\cap B))".
From the latter we find that: "0.53-0.45=P(B)-P(A\\cap B)" . We substitute "P(B)=0.22" and get: "P(A\\cap B)=0.22-0.08=0.14".
Denote the whole probability space by "X". Then, "\\bar{A}=X\\backslash A" and "\\bar{B}=X\\backslash B". Thus, we get: "\\bar{A}\\cup{\\bar{B}}=(X\\backslash A)\\cup(X\\backslash B)=X\\backslash(A\\cap B)". We get:
a). "P(\\bar{A}\\cup{\\bar{B}})=P(X\\backslash(A\\cap B))=P(X)-P(A\\cap B)=1-0.14=0.86".
b). For independent events one has: "P(A\\cap B)=P(A)P(B)". "P(A)P(B)=0.099". It follows from the formulation of the problem. As we can see "P(A)P(B)\\neq P(A\\cap B)". Thus, the events are not independent.
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