Zadanie 1d, Paper 3, maj 2024, TZ1, Matematyka AA IB HL

A production line compares two randomized inspection policies. Under policy A, each batch is inspected independently with constant probability p, where 0<p<1; X is the number of batches up to and including the first inspection. You may use $\operatorname{Var}(X)=(1-p)/p^2$. Under policy B, if the preceding j−1 batches since the last inspection were not inspected, the conditional probability of inspecting batch j is q_j = j²/16 for j=1,2,3,4. After an inspection j resets to 1. Y is the number of batches until the first inspection under B.