Start by clarifying the assumptions: each user independently uses the feature with probability p, and we want the probability that at least one user uses it. Use the complement rule: P(at least one) = 1 - P(none), where P(none) = (1-p)^n. Then discuss implications for product analytics, such as expected usage and sample size calculations.
Pro tip: Connect the formula to practical scenarios: for example, if p is small and n is large, the probability can be approximated by 1 - e^{-np}, which is useful for quick estimates. Also, mention that this is a common building block for more complex models like binomial distributions.
Confirm that each user's usage is independent and identically distributed with probability p. Ensure the question asks for at least one user, not exactly one.
The complement of 'at least one user uses the feature' is 'no user uses the feature'. Calculate the probability of this event.
Since users are independent, the probability that none use the feature is (1-p)^n.
Subtract from 1 to get the desired probability: P(at least one) = 1 - (1-p)^n.
Discuss what this means for product metrics, such as expected number of users, and mention extensions like binomial distribution or Poisson approximation for large n and small p.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the setup: assume N independent users, each with probability p of using the feature. The question asks for P(user 1 used | at least one user used). Use Bayes' theorem or the complement rule to compute this conditional probability, then simplify the expression and discuss its behavior as N and p vary.
Pro tip: Always state your assumptions explicitly (independence, identical p) and then discuss how the result changes if those assumptions are violated—this shows you understand the model's limitations and can think critically about real-world data.
State that there are N users, each independently uses the feature with probability p. Let A be the event that user 1 used the feature, and B be the event that at least one user used it.
Write P(A|B) = P(A ∩ B) / P(B). Since A implies B (if user 1 used it, then at least one user did), P(A ∩ B) = P(A) = p.
P(B) = 1 - P(no one used) = 1 - (1-p)^N, assuming independence.
Thus P(A|B) = p / [1 - (1-p)^N]. Discuss special cases: as N→∞, this approaches p (if p>0); as p→0, it approaches 1/N; and if N=1, it equals 1.
Explain how this probability informs feature adoption metrics, such as the likelihood that a specific user is an adopter given that the feature is used at all, and discuss implications for A/B testing or user segmentation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the underlying distribution of users using the feature (e.g., Poisson or binomial) and the total number of users. Then, apply the formula for the conditional expectation of a truncated distribution, such as E[X | X ≥ 1] = E[X] / (1 - P(X=0)) for Poisson, and compute the numerical answer. Finally, interpret the result in the context of Coinbase's product analytics, discussing implications for feature adoption and A/B testing.
Pro tip: Always state your assumptions explicitly and check if the interviewer expects a specific distribution; in practice, you might need to estimate parameters from data, so mention how you would validate the model.
Ask or state the assumed distribution of users (e.g., Poisson, binomial) and the total number of users or the rate parameter. Confirm whether users are independent and if the feature usage is rare.
Let X be the number of users who used the feature. We want E[X | X ≥ 1]. Write the conditional expectation formula: E[X | X ≥ 1] = E[X] / P(X ≥ 1) for non-negative integer X.
For Poisson(λ), E[X] = λ and P(X ≥ 1) = 1 - e^{-λ}, so E[X | X ≥ 1] = λ / (1 - e^{-λ}). For binomial(n, p), E[X] = np and P(X ≥ 1) = 1 - (1-p)^n, so E[X | X ≥ 1] = np / (1 - (1-p)^n). Plug in the given numbers.
Explain what the result means for Coinbase: e.g., if the expected number is high, the feature is likely used by multiple users; if low, it may be niche. Relate to A/B testing by noting that conditioning on at least one user can affect metrics like average usage per user.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.