Compute wallet-link probabilities and conditionals
Overview
This question evaluates understanding of Bernoulli trials, binomial probabilities, expectation, conditional probabilities, and dependence bounds (Fréchet bounds) within the domain of probability theory and statistics.
Coinbase
Oct 13, 2025, 9:49 PM
Data Scientist
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Statistics & Math
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Wallet-Linking Probability Exercises
Context: Each user links a wallet independently with probability p. Treat each user as a Bernoulli trial with success = "links a wallet."
Tasks
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For n users with identical independent probability p:
- Derive P(at least one links) and P(exactly k link).
- Provide closed forms and discuss edge cases: p = 0, p = 1, large n, and the small-p Poisson approximation.
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Evaluate numerically for p = 0.30 and n = 5:
- P(at least one)
- P(exactly 2)
- Expected number of links
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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)
- P(A ∩ B | A ∪ B)
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Without assuming independence but given P(A ∪ B) = 0.70 with marginals p_A = 0.40 and p_B = 0.60:
- State the feasible range for P(A ∩ B) using Fréchet bounds and the corresponding range for P(A ∩ B | A ∪ B).
- Explain when each boundary is attained and what dependence structure it implies.
Solution
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