What statement is true concerning the Gini index and classification error rate?

The Gini index and the classification error rate are both key measures used in evaluating the performance of classification models, particularly in the context of binary classification. The statement that if p1 equals p2, they are equal is true because both the Gini index and the classification error rate are dependent on the probability distributions of the predicted classes.

When p1 (the probability of the positive class) is equal to p2 (the probability of the negative class), this indicates a situation where the model does not discriminate between the classes—it essentially has no predictive power. In this case, the Gini index, which measures inequality or purity in class distribution, will yield a value of 0, signifying that there is no separation between the classes. Similarly, the classification error rate, which represents the proportion of misclassified instances, would also be at its maximum value, corresponding to a model with no predictive ability.

Thus, when both probabilities are equal, the Gini index reflects this lack of discrimination by also reaching a baseline of 0, aligning it with the classification error in this specific scenario. This shared behavior emphasizes the relationship between model performance metrics like the Gini index and classification error rate under conditions of equal class probabilities.