Two events R and S are said to be independent if the occurence of R Makes it neither more nor less probable that S occurs and , conversely, if the occurrence of R makes it neither more nor less prbable that R occurs.
This implies that if; P(Sophomore|Robotics)= P(Sophomore), then they and independent, conversly if they are not equal, they are dependent.
[tex]\begin{gathered} p(sophomore|robotics)=\frac{p(S\cap R)}{p(R)}=\frac{\frac{18}{76}}{\frac{39}{76}} \\ =\frac{18}{39}=\frac{6}{13} \\ P(sophomore)=\frac{38}{76}=\frac{1}{2} \\ \sin ce, \\ p\mleft(sophomore|robotics\mright)\ne P(sophomore) \\ \text{Therefore they are dependent because }p(sophomore|robotics)=\frac{6}{13}\ne P(sophomore)=\frac{1}{2} \end{gathered}[/tex]