Mathematics - Probability Question with Solution | TestHub

Since, $P($ exactly one of $A,B$ occurs $)=q$. $\therefore P(A\cup B)-P(A\cap B)=q$ $\Rightarrow p-P(A\cap B)=q\Rightarrow P(A\cap B)=p-q$ $\Rightarrow 1-P\left(A^{\prime }\cup B^{\prime }\right)=p-q\Rightarrow P\left(A^{\prime }\cup B^{\prime }\right)=1-p+q$ $\Rightarrow P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-P\left(A^{\prime }\cap B^{\prime }\right)=1-p+q$ $\Rightarrow P\left(A^{\prime }\right)+P\left(B^{\prime }\right)=(1-p+q)+[1-P(A\cup B)]$ $=(1-p+q)+(1-p)=2-2p+q$