Compute wallet-link probabilities and conditionals

Assume independent users link a wallet with probability p. 1) For n users, derive P(at least one links) and P(exactly k link). Provide closed forms and edge cases (p=0, p=1, large n, small p Poisson approximation). 2) Evaluate numerically for p=0.30 and n=5: P(at least one), P(exactly 2), and the expected number of links. 3) For two specific 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). 4) 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.