Comparing Two Ad Insertion Methods: 4% Random vs Fixed 1-in-25

Context

You are designing an ad insertion system. In any short time bucket or session, suppose there are n eligible content slots (impressions) where an ad could be inserted.

Two teams propose different policies:

• Team A: Replace each slot independently with probability p = 0.04 (4% random replacement).
• Team B: Use a fixed periodic schedule of exactly 1 ad every 25 slots (a 1-in-25 slot), i.e., a deterministic pattern with period 25.

Assume buckets are contiguous sequences of n slots; when relevant, assume the starting phase of the fixed schedule is uniformly random across the 25 positions (this models arbitrary bucket alignment in production).

Task

1. For each method, compute in a bucket of n slots:
   • Expected number of ads.
   • Variance of the number of ads.
2. For each method, compute (or tightly bound/approximate) the probability that the number of ads in a bucket exceeds twice its expectation (risk of overload).
3. Briefly compare the methods in terms of expected counts, variability (load smoothing), and overload risk.

Hint: For the 4% random method, model the count of ads with a binomial distribution.