question_answer

If A, B and C are exhaustive events satisfying \[P\,\,((A\cup B)\cap \overline{C}=\frac{1}{5},\]\[P\,\,(B\,\cap \,C)-P\,\,(A\,\cap \,B\,\cap \,C)=\frac{1}{15}\] and \[P\,\,(A\,\cap \,C)\frac{1}{10},\] then \[P\,\,(C\,\cap \,\,(A\,\cup \,B))\] equal to-

A) \[\frac{17}{30}\]

B)  \[\frac{18}{30}\]  

C) \[\frac{19}{30}\]           

D) \[\frac{20}{30}\]

Correct Answer: C

Solution :

\[\frac{1}{5}=x+v+y\]

\[\frac{1}{15}=w\]

\[\frac{1}{10}=u+t\]

\[x+y+z+u+v+w+t=1\]

\[z=1-\frac{1}{5}-\frac{1}{15}-\frac{1}{10}=\frac{19}{30}\]