Video Recommendation Push: Selection Probabilities, Complements, and Design Choices

Scenario

You are designing a push-notification system that recommends k videos to a user from a large catalog of N videos. You also need to reason about whether to push the same or different videos to two friends and how to model this recommendation problem.

Assumption (unless otherwise stated): The system selects k distinct videos without replacement. If order is relevant (e.g., top-1, top-2, …), we treat the result as an ordered list; otherwise as an unordered set.

Questions

a) From an inventory of N videos, what is the probability that a specific ordered set of k distinct videos is pushed to a user? What about any specific unordered subset of size k?

b) Given an event’s probability p, compute its complementary probability and apply it to the video-selection context (e.g., the probability that at least one of a set of m target videos appears in the k selected videos).

c) For two friends, should we push the same video to both or different videos? Discuss pros, cons, and expected impact on metrics.

d) Which statistical or machine-learning model(s) would you use for this recommendation problem and why?