A) \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})=P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\]
B) \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})>P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\]
C) \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})\le P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\]
D) None of these
Correct Answer: C
Solution :
For any two events \[A\] and \[B,\]we have \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] \[\therefore \,\,\,P(A\cup B)\le P(A)+P(B).\] Using principle of mathematical induction, it can be easily established that \[P\left( \underset{i=1}{\overset{n}{\mathop{\cup }}}\,{{A}_{i}} \right)\le \sum\limits_{i=1}^{n}{P({{A}_{i}}).}\]
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