a$.$ Please compute the $($i$)$ entropy; $($ii$)$ Gini; $($iii$)$ classification error of cases $(1)$ and $(2).$ Assume they are data at a node of the decision tree. Note $$C$0$ and $$C$1$ are two classes, and the number after the colon indicates the number of samples in each class.

$(1)$ C$0$: $8$ and C$1$: $12$

$(2)$ C$0$: $6$ and C$1$: $14$

b$.$ Assume we plan to split the data for cases $(1)$ and $(2)$ in the following ways. Could you calculate the information gain for each? Do you suggest a split based on these, and why?$$

Hint: the definition of $$information gain$$ can be found on page $45$ of Lecture$-6.$

$(1)$ C$0$: $1$ and C$1$: $9+$ C$0$: $7$ and C$1$: $3$

$(2)$ C$0$: $3$ and C$1$: $3+$ C$0$: $3$ and C$1$: $11$