grandes-ecoles 2022 Q22

grandes-ecoles · France · centrale-maths2__mp Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), explain how to deduce the formula $$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$ 
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), explain how to deduce the formula
$$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$
Paper Questions
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