Derive Probability of Even Sum in Bernoulli Trials
Overview
The question evaluates probabilistic reasoning and discrete stochastic-process concepts, specifically parity of sums of i.i.d. Bernoulli(p) random variables and expected waiting times for consecutive outcomes in fair coin flips.
Boston Consulting Group
Aug 4, 2025, 10:55 AM
Data Scientist
Take-home Project
Statistics & Math
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Probability Puzzles: Parity and Runs
Context
You are given two independent probability problems commonly seen in data-science take-home assessments. Assume all random variables are defined on the same probability space.
Problems
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Parity of a Bernoulli Sum
- Let X1, ..., Xn be i.i.d. Bernoulli(p). Let S = Σ_{i=1}^n X_i.
- Derive a closed-form expression for P(S is even).
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Waiting Time to First Repeat (Fair Coin)
- A fair coin is flipped until two consecutive identical outcomes (HH or TT) appear.
- Find the expected number of flips required.
Solution
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