Independence with three or more events

A question is this type if and only if it involves three or more events where pairwise or mutual independence is stated or must be verified, often with Venn diagrams.

2 questions

OCR S4 2017 June Q3
3 For events \(A , B\) and \(C\) it is given that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5 , \mathrm { P } ( C ) = 0.4\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.1\). It is also given that events \(A\) and \(B\) are independent and that events \(A\) and \(C\) are independent.
  1. Find \(\mathrm { P } ( B \mid A )\).
  2. Given also that events \(B\) and \(C\) are independent, find \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
  3. Given instead that events \(B\) and \(C\) are not independent, find the greatest and least possible values of \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
Edexcel S1 2004 June Q6
6. Three events \(A , B\) and \(C\) are defined in the sample space \(S\). The events \(A\) and \(B\) are mutually exclusive and \(A\) and \(C\) are independent.
  1. Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that \(\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup C ) = 0.7\), find
  2. \(\mathrm { P } ( A C )\),
  3. \(\mathrm { P } ( A \cup B )\),
  4. \(\mathrm { P } ( C )\). END