Click Probability Across Repeated Impressions

Context: We show A impressions of the same item to a user. Unless otherwise stated, each impression is an independent Bernoulli trial with click probability p.

1. Constant p

• Derive P(no clicks after A impressions) as a function of A and p.
• For p = 0.3, explain why the correct expression for the y-axis labeled P(no clicks) is (1 − 0.3)^A rather than 0.3^A.
• Describe qualitatively how this curve behaves as A increases.

2. Time-to-first-click (Geometric)

• Generalize to arbitrary p and compute the expected number of impressions until the first click (i.e., the geometric distribution result).

3. Heterogeneous propensities (Beta prior)

• Suppose the user-level click propensity p varies by user with prior p ~ Beta(α, β). Derive the marginal probability of no clicks after A impressions and express it using Beta functions (i.e., the Beta–Binomial model).
• Interpret how heterogeneity changes the tail behavior versus the i.i.d. fixed-p case.

4. Fatigue: decaying click probability

• Suppose p decays with impression index due to fatigue: p_k = p_0 · e^{−λ(k−1)} for k = 1, 2, ..., A.
• Provide an expression (or tight bounds) for P(no clicks after A impressions).
• Briefly discuss how you would estimate λ from data.