##### Scenario When users sign up for Coinbase, each of the `n` users is independently prompted to link a crypto wallet. Each user links their wallet with probability `p` (and fails to link with probability `1 − p`), independently of every other user. ##### Question Using this model, answer the following: 1. What is the expected number of users who link their wallet? 2. What is the probability that at least one of the `n` users links a wallet? 3. For two specific users A and B, given that at least one of A or B links a wallet, what is the probability that **both** A and B link their wallets? ##### Hints - Part 1: Use linearity of expectation with indicator variables — you do not even need independence here. - Part 2: Use the complement rule; it is easier to compute the probability that *no one* links. - Part 3: Use the definition of conditional probability, `P(A ∩ B | A ∪ B) = P(A ∩ B) / P(A ∪ B)`.

Coinbase probability question covering Bernoulli indicators, expected wallet links, complement rule for at least one success, and conditional probability for two users linking wallets.

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What difficulty level is this interview question?

This is a easy difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Coinbase.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Coinbase during technical interviews.

Calculate Expected Users Linking Coinbase Wallets

Expected Coinbase Wallet Linking

When users sign up for Coinbase, each of the n users is independently prompted to link a crypto wallet. Each user links their wallet with probability p and fails to link with probability 1 - p, independently of every other user.

Constraints & Assumptions

Use the Bernoulli/binomial model described above.
State where independence is required and where it is not.
Keep answers symbolic in terms of n and p .
For the two-user conditional probability, let A be "user A links" and B be "user B links."

Clarifying Questions to Ask

Are all users assumed to have the same wallet-linking probability p ?
Are linking decisions independent across users?
Is the interviewer asking for expected value, probability, or conditional probability?

Part 1 - Expected Number of Linked Wallets

What is the expected number of users who link their wallet?

What This Part Should Cover

Define indicator variables for each user.
Use linearity of expectation.
Show that the expected count is n * p .
Note that independence is not required for this expectation if each user has success probability p .

Part 2 - Probability at Least One User Links

What is the probability that at least one of the n users links a wallet?

What This Part Should Cover

Use the complement rule.
Compute the probability that no users link as (1 - p)^n .
Conclude that the probability at least one links is 1 - (1 - p)^n .
State that independence is needed for the product form.

Part 3 - Conditional Probability for Two Users

For two specific users A and B, given that at least one of A or B links a wallet, what is the probability that both A and B link their wallets?

What This Part Should Cover

Use P(A and B | A or B) = P(A and B) / P(A or B) .
Compute P(A and B) = p^2 .
Compute P(A or B) = 2p - p^2 .
Simplify to p / (2 - p) for p > 0 .

What a Strong Answer Covers

A strong answer distinguishes expected counts from probabilities, applies complement and conditional-probability rules correctly, and states when independence matters.

Follow-up Questions

What changes if each user has their own probability p_i ?
What is the variance of the number of users who link?
What happens to P(at least one) as n gets large?

Expected Coinbase Wallet Linking

Constraints & Assumptions

Use the Bernoulli/binomial model described above.
State where independence is required and where it is not.
Keep answers symbolic in terms of n and p .
For the two-user conditional probability, let A be "user A links" and B be "user B links."

Clarifying Questions to Ask

Are all users assumed to have the same wallet-linking probability p ?
Are linking decisions independent across users?
Is the interviewer asking for expected value, probability, or conditional probability?

Part 1 - Expected Number of Linked Wallets

What is the expected number of users who link their wallet?

What This Part Should Cover

Define indicator variables for each user.
Use linearity of expectation.
Show that the expected count is n * p .
Note that independence is not required for this expectation if each user has success probability p .

Part 2 - Probability at Least One User Links

What is the probability that at least one of the n users links a wallet?

What This Part Should Cover

Use the complement rule.
Compute the probability that no users link as (1 - p)^n .
Conclude that the probability at least one links is 1 - (1 - p)^n .
State that independence is needed for the product form.

Part 3 - Conditional Probability for Two Users

For two specific users A and B, given that at least one of A or B links a wallet, what is the probability that both A and B link their wallets?

What This Part Should Cover

Use P(A and B | A or B) = P(A and B) / P(A or B) .
Compute P(A and B) = p^2 .
Compute P(A or B) = 2p - p^2 .
Simplify to p / (2 - p) for p > 0 .

What a Strong Answer Covers

A strong answer distinguishes expected counts from probabilities, applies complement and conditional-probability rules correctly, and states when independence matters.

Follow-up Questions

What changes if each user has their own probability p_i ?
What is the variance of the number of users who link?
What happens to P(at least one) as n gets large?

Calculate Expected Users Linking Coinbase Wallets

Quick Overview

Calculate Expected Users Linking Coinbase Wallets

Expected Coinbase Wallet Linking

Constraints & Assumptions

Clarifying Questions to Ask

Part 1 - Expected Number of Linked Wallets

What This Part Should Cover

Part 2 - Probability at Least One User Links

What This Part Should Cover

Part 3 - Conditional Probability for Two Users

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

Write your answer

Calculate Expected Users Linking Coinbase Wallets

Quick Overview

Calculate Expected Users Linking Coinbase Wallets

Expected Coinbase Wallet Linking

Constraints & Assumptions

Clarifying Questions to Ask

Part 1 - Expected Number of Linked Wallets

What This Part Should Cover

Part 2 - Probability at Least One User Links

What This Part Should Cover

Part 3 - Conditional Probability for Two Users

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

Write your answer