Derive Probability of Even Sum in Bernoulli Trials
Company: Boston Consulting Group
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Take-home Project
##### Scenario
Online assessment requires candidates to solve non-formulaic probability puzzles within strict proctoring constraints.
##### Question
Let X1,…,Xn be i.i.d. Bernoulli(p). Derive a closed-form expression for P(Σ_i X_i is even). A fair coin is flipped until two consecutive identical outcomes appear. What is the expected number of flips?
##### Hints
Use moment-generating tricks or parity arguments for the first; write a recurrence or Markov chain for the second.
Quick Answer: 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.