Question: Let S be the sample space.

(a) What do you mean when we say that the collection of events \({{A}_{1}},....,{{A}_{n}}\) is mutually exclusive and collectively partition of S?

(b) Suppose that the events \({{A}_{1}},....,{{A}_{n}}\) form a mutually exclusive and collectively exhaustive partition of S, and let \(B\subset S\). Justify each line in the following set of equations:

\[\Pr \left( B \right)=\Pr \left( B\cap S \right)=\Pr \left( B\cap \left( {{A}_{1}}\cup ....\cup {{A}_{n}} \right) \right)=\Pr \left( \left( B\cap {{A}_{1}} \right)\cup ...\cup \left( B\cap {{A}_{n}} \right) \right)\] \[=\Pr \left( B\cap {{A}_{1}} \right)+.....+\Pr \left( B\cap {{A}_{n}} \right)=\Pr \left( B|{{A}_{1}} \right)\Pr \left( {{A}_{1}} \right)+....+\Pr \left( B|{{A}_{n}} \right)\Pr \left( {{A}_{n}} \right)\]