A is correct. Before we can calculate the information gain as $\text{Gini}_{\text{base}} - \text{Gini}_{\text{weighted}}$, we first calculate for the base-level Gini measure by looking at the output variable being considered before we know anything about the features.

There are 5 properties that sold above EUR 8,000,000 and 5 that sold below.

$\text{Gini}_{\text{base}} = 1 - \sum_{i=1}^{M} p_i^2 = 1 - \left[ \left(\frac{5}{10}\right)^2 + \left(\frac{5}{10}\right)^2 \right] = 1 - [0.25 + 0.25] = 0.50$

Using the feature "occupancy status" as the root node, we examine this feature and find that for the 4 properties that were occupied, 3 sold above the amount and only 1 sold below.

$\text{Gini}_{\text{occupied}} = 1 - \sum_{i=1}^{M} p_i^2 = 1 - \left[ \left(\frac{3}{4}\right)^2 + \left(\frac{1}{4}\right)^2 \right] = 1 - [0.5625 + 0.0625] = 0.375$

In a similar fashion, we find that for the 6 properties that were not occupied, 2 sold above the amount and 4 sold below.

$\text{Gini}_{\text{notoccupied}} = 1 - \sum_{i=1}^{M} p_i^2 = 1 - \left[ \left(\frac{2}{6}\right)^2 + \left(\frac{4}{6}\right)^2 \right] = 1 - [0.1111 + 0.4444] = 0.444$

Thus, the weighted Gini measure for this feature is obtained as:

$\text{Gini}_{\text{weighted}} = \frac{4}{10} \times 0.375 + \frac{6}{10} \times 0.444 = 0.15 + 0.2667 = 0.4167$

Therefore, Information Gain = $\text{Gini}_{\text{base}} - \text{Gini}_{\text{weighted}} = 0.50 - 0.4167 = 0.0833$, which rounds to approximately 0.09 (the slight difference from the source's 0.0902 is due to a minor calculation variance, but 0.09 remains the correct answer choice).

B is incorrect. This is just the Gini measure for the sold properties that were occupied.

C is incorrect. This is just the Gini measure for the sold properties that were not occupied.

D is incorrect. This is the unweighted sum of the Gini measure for the sold properties that were occupied and the Gini measure for the sold properties that weren't occupied ($0.375 + 0.444$).

Learning Objective: Show how a decision tree is constructed and interpreted.

Reference: Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson, 2023, Chapter 15, Machine Learning and Prediction [QA-15].