##### Question Coinbase users each independently link a wallet with probability p. Work through the full probability model for this process. 1. **General formulas and edge cases.** For n independent users, derive P(at least one user links) and P(exactly k users link). Give closed forms, and state behavior at the edge cases p = 0, p = 1, large n, and the small-p / large-n Poisson approximation. 2. **Expectation.** Derive the expected number of users who link a wallet, stating the independence assumption clearly (and noting where it is and is not required). 3. **Poisson approximation with error term.** For small p and large n, give the Poisson approximation for P(at least one links), then derive its first-order error term (how far the approximation is from the exact value, and in which direction). 4. **Numerical evaluation.** For p = 0.30 and n = 5, compute P(at least one), P(exactly 2), and the expected number of links. 5. **Two specific users, independent.** For two users A and B with independent probabilities p_A = 0.40 and p_B = 0.60, compute P(A ∪ B), P(A ∩ B), and P(A ∩ B | A ∪ B). 6. **Two users, dependence unknown.** If independence is not assumed but you are told P(A ∪ B) = 0.70 with marginals p_A = 0.40, p_B = 0.60, give the feasible range for P(A ∩ B) (Fréchet bounds) and the corresponding range for P(A ∩ B | A ∪ B). Explain when each boundary is attained and what dependence structure it implies.