We must determine whether these occurrences are pairwise independent, as well as whether all three events are mutually independent. The following definitions and propositions are required.
If for two events \({\rm{A}}\)and \({\rm{B}}\)stands
\({\rm{P(A}}\mid {\rm{B) = P(A)}}\)
We call them self-sufficient. Otherwise, they are reliant.
Property of Multiplication: Two events If and only if, \({\rm{A}}\) and \({\rm{B}}\) are independent.
\({\rm{P(A}} \cap {\rm{B) = P(A)*P(B)}}\)
Multiplication Property:
For events \({{\rm{A}}_{\rm{1}}}{\rm{,}}{{\rm{A}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{A}}_{\rm{n}}}{\rm{,n}} \in {\rm{N}}\) we say that they are mutually independent if
\({\rm{P}}\left( {{A_{{i_1}}} \cap {A_{{i_2}}} \ldots {A_{{i_k}}}} \right){\rm{ = P}}\left( {{{\rm{A}}_{{{\rm{i}}_{\rm{1}}}}}} \right){\rm{*P}}\left( {{{\rm{A}}_{{{\rm{i}}_{\rm{2}}}}}} \right){\rm{* \ldots *P}}\left( {{{\rm{A}}_{{{\rm{i}}_{\rm{k}}}}}} \right)\)
for every \({\rm{k}} \in {\rm{\{ 2,3, \ldots ,n\} }}\), and every subset of indices \({{\rm{i}}_{\rm{1}}}{\rm{,}}{{\rm{i}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{i}}_{\rm{k}}}\). First, notice the following intersections.
\({\rm{A}} \cap {\rm{B = \{ (3,4)\} }}\)
\(\begin{array}{c}A \cap {\rm{B = \{ (3,4)\} }};\\A \cap {\rm{C = \{ (3,4)\} }};\\B \cap {\rm{C = \{ (3,4)\} }};\\A \cap B \cap {\rm{C = \{ (3,4)\} }}.\end{array}\)
We have that
\(\begin{array}{c}{\rm{P(A}} \cap {\rm{B) = P(A}} \cap C)\\ = P(B \cap {\rm{C) = P(A}} \cap B \cap {\rm{C) = }}\frac{{\rm{1}}}{{{\rm{36}}}}\end{array}\)
The following is true
\(\begin{array}{l}{\rm{P(A)*P(B) = }}\frac{{\rm{1}}}{{\rm{6}}}*\frac{{\rm{1}}}{{\rm{6}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{ = }}P(A \cap B)\\{\rm{P(A)*P(C) = }}\frac{{\rm{1}}}{{\rm{6}}}*\frac{{\rm{1}}}{{\rm{6}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{ = }}P(A \cap C)\\{\rm{P(B)*P(C) = }}\frac{{\rm{1}}}{{\rm{6}}}*\frac{{\rm{1}}}{{\rm{6}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{ = }}P(B \cap C)\end{array}\)
