m m m (i) (a) If p1 +P2 +...+Pm=1, prove the identity { pk p; =1- pk. k=1 j=1,ji k=1 (b) Explain how this identity can be used to calculate a measure of the effectiveness of a decision tree. A researcher is considering two possible decision trees to classify items of four different types labelled A, B, C and D. A sample of 15 items classified using each of the trees gave the results shown below. YES YES TEST 1 AAAB TEST 4 BBBBD NO NO YES YES TEST 2 BCCC TEST 5 AAAC NO NO YES YES TEST 3 DDDD TEST 6 CC NO NO BBB BDDD TREE 1 TREE 2 (ii) (a) Calculate the Gini index for each tree. (b) Comment on your answers to part (ii)(a)