Compute wallet-link probabilities, expectation, and conditionals

Question

Each Coinbase user links a wallet independently with probability $p$. Work through the complete probability model for this process.

1. General formulas and edge cases. For $n$ independent users, derive $P(\text{at least one user links})$ and $P(\text{exactly } k \text{ users link})$. Provide closed forms, and describe the 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, making the independence assumption explicit (and noting where it is needed and where it is not).

3. Poisson approximation with error term. For small $p$ and large $n$, give the Poisson approximation for $P(\text{at least one links})$, 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 $P(\text{at least one})$, $P(\text{exactly } 2)$, 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 $P(A \cup B)$, $P(A \cap B)$, and $P(A \cap B \mid A \cup B)$.

6. Two users, dependence unknown. If independence is not assumed but you are told $P(A \cup B) = 0.70$ with marginals $p_A = 0.40$, $p_B = 0.60$, give the feasible range for $P(A \cap B)$ (Fréchet bounds) and the corresponding range for $P(A \cap B \mid A \cup B)$. Explain when each boundary is attained and what dependence structure it implies.

Overview: A Coinbase data science probability question about independent wallet-link events: derive the expected number of linkers and $P(\text{at least one links})$, give the Poisson approximation with its first-order error term, and extend to binomial probabilities, conditional probabilities, and Fréchet dependence bounds. Includes worked numeric examples.