Solve Classic Probability Questions
Quick Overview
This set evaluates probabilistic reasoning and expectation computation across topics such as sequential conditional survival, convolution of continuous distributions, coupon-collector expectations, variance under uniform sampling in a ball, and constrained online selection decisions, and falls under Statistics & Math with emphasis on probability theory and stochastic processes. These problems are commonly asked to probe analytical probability intuition, facility with integration and combinatorial expectation, handling of information updates and dependence, and formulation of online decision rules; the level of abstraction spans both conceptual understanding and practical application requiring formal probabilistic derivations.
An onsite interview included the following independent probability and expectation questions:
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Squid Game bridge problem.
In the classic glass-bridge setup, there are
npositions to cross. At each position, exactly one of two panels is safe and the other breaks. Players cross sequentially. Once a panel is tested, later players know which panel is safe at that position. If you are them-th player in line, what is your probability of surviving? -
Sum of two uniforms.
Let
X, Y ~ Uniform(0,1)independently. DefineS = X + Y. What is the PDF ofS? - Collect all die faces. You repeatedly roll a fair 6-sided die. What is the expected number of rolls needed to see all 6 faces at least once?
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Variance of one coordinate in a 3D unit ball.
A point is sampled uniformly from the
solid
3D unit ball
B = {(x,y,z): x^2 + y^2 + z^2 <= 1}. If the sampled point is(X,Y,Z), what isVar(X)? -
Online selection from uniform draws.
You observe
ni.i.d. draws fromUniform(0,1)one at a time. After seeing each draw, you must irrevocably decide whether to keep it or discard it. By the end, you must keep exactlykdraws. What is the optimal strategy, and how would you compute the optimal expected total value of the kept draws? (If the interviewer instead asks for expected average value, that is just the expected total divided byk.)