1. Prove that Equation (5.2) suffices for showing that P[LD(A(S))≥ 1/8] ≥ 1/7.

Hint: Let θ be a random variable that receives values in [0, 1] and whose expectation satisfies E[θ] ≥ 1/4. Use Lemma B.1 to show that P[θ ≥ 1/8] ≥ 1/7.

2. Assume you are asked to design a learning algorithm to predict whether patients are going to suffer a heart attack. Relevant patient features the algorithm may have access to include blood pressure (BP), body-mass index (BMI), age (A), level of physical activity (P), and income (I).

You have to choose between two algorithms; the first picks an axis aligned rectangle in the two dimensional space spanned by the features BP and BMI and the other picks an axis aligned rectangle in the five dimensional space spanned by all the preceding features.