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[See Steps] Let S be the sample space. What do you mean when we say that the collection of events A_1,....,A_n is mutually exclusive and collectively partition

Question: Let S be the sample space.

  1. What do you mean when we say that the collection of events \({{A}_{1}},....,{{A}_{n}}\) is mutually exclusive and collectively partition of S ?
  2. 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:
\[\Pr \left( B \right)=\Pr \left( B\cap S \right)=\Pr \left( B\cap \left( {{A}_{1}}\cup ....\cup {{A}_{n}} \right) \right)=\Pr \left( \left( B\cap {{A}_{1}} \right)\cup ...\cup \left( B\cap {{A}_{n}} \right) \right)\] \[=\Pr \left( B\cap {{A}_{1}} \right)+.....+\Pr \left( B\cap {{A}_{n}} \right)=\Pr \left( B|{{A}_{1}} \right)\Pr \left( {{A}_{1}} \right)+....+\Pr \left( B|{{A}_{n}} \right)\Pr \left( {{A}_{n}} \right)\]

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