Recognize that each user's wallet linking is an independent Bernoulli trial with success probability p. The total number of linked users is the sum of n independent Bernoulli variables, so the expected value is n * p. State the result clearly and explain the reasoning.
Pro tip: Mention that this is a binomial distribution and that the expectation is linear, so you don't need independence for the expectation to hold—though independence is given here. This shows you understand the underlying assumptions and can generalize.
Let X_i be an indicator variable that equals 1 if user i links their wallet, and 0 otherwise. Then X_i ~ Bernoulli(p).
The total number of linked users is S = X_1 + X_2 + ... + X_n.
Since expectation is linear, E[S] = E[X_1] + E[X_2] + ... + E[X_n] = n * p.
Thus, the expected number is n * p. Also note that S follows a Binomial(n, p) distribution, so the variance is n * p * (1-p).
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Complement rule, so 1 minus the probability nobody links, which is 1 minus (1-p)^n.
Clarify the assumptions about independence and the probability of a single user linking their wallet. Then model the number of users who link as a binomial random variable and compute the probability of at least one success as 1 minus the probability of zero successes. If the per-user probability is not given, denote it as p and express the answer in terms of p and n.
Pro tip: Always state your assumptions explicitly and consider edge cases like p=0 or p=1; also mention that in real product analytics, user behaviors are often not independent, so you might need to account for correlation or use historical data to estimate p.
Ask clarifying questions to confirm whether each user links independently and whether the probability of linking is the same for all users. If not provided, denote the probability as p.
Let X be the number of users who link their wallet out of n. Under the assumptions, X follows a Binomial(n, p) distribution.
Use the complement rule: P(X ≥ 1) = 1 - P(X = 0) = 1 - (1 - p)^n.
Consider what happens if p is very small or n is large, and mention that if p is unknown, it must be estimated from data. Also note that independence may not hold in practice.
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Clarify the assumptions about the joint distribution of wallet linking between users A and B, then apply conditional probability. If independence is assumed, compute P(A and B) / P(A or B) using given or estimated marginal probabilities; otherwise, discuss how correlation or additional data would be needed.
Pro tip: In real-world product analytics, user behaviors are rarely independent; always check for correlation (e.g., via historical data or cohort analysis) before assuming independence. Mentioning this shows you understand practical data nuances.
Let A be the event that user A links a wallet, B for user B. The condition is that at least one links, i.e., A ∪ B. We want P(A ∩ B | A ∪ B).
Use P(A ∩ B | A ∪ B) = P(A ∩ B) / P(A ∪ B). Expand P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
If A and B are independent, P(A ∩ B) = P(A)P(B). Then the probability simplifies to P(A)P(B) / (P(A)+P(B)-P(A)P(B)). If not independent, you need the joint probability or correlation.
Use historical data to estimate P(A), P(B), and possibly P(A ∩ B). If only marginal probabilities are available, state the independence assumption explicitly and compute.
Present the computed probability and discuss how violations of independence (e.g., due to referral, shared device, or temporal factors) could affect the result.
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