Derive no-click probability and sketch implications

Each impression is independently clicked with probability p. 1) For A impressions, derive P(no clicks) as a function of A and p, and explain why (for p=0.3) it is (1−0.3)^A rather than 0.3^A when the y-axis is P(no clicks). Describe qualitatively how the curve behaves as A increases. 2) Generalize to arbitrary p and compute the expected number of impressions until the first click (geometric distribution). 3) If user-level click propensities vary with a Beta(α,β) prior (heterogeneity), derive the marginal probability of no clicks after A impressions and express it using Beta functions (Beta–Binomial), then interpret how heterogeneity changes the tail vs. the i.i.d. case. 4) Suppose p decays with impression index due to fatigue, e.g., p_k = p_0·e^{−λ(k−1)}. Provide an expression (or tight bound) for P(no clicks after A impressions) and discuss how you would estimate λ from data.