Marbles Tricky Probability Puzzle

This puzzle is about arranging a fixed mix of "good" and "bad" outcomes between two choices when you know you will later pick one uniformly at random and then sample once from it. The concrete story uses colored marbles and jars, but the structure is general to many quant settings: you control the composition of two risky buckets, then face a blind, single-shot draw. The candidate must reason about how to place the favorable items across the two containers to maximize the overall chance of success, and to see why the obvious symmetric or "fair" splits are not necessarily optimal.

The question leans on basic probability, expected value over discrete scenarios, and simple optimization under constraints. It tests whether the candidate can express the overall success probability as a function of the allocation decision and then think globally about that function's shape, rather than just trying ad hoc distributions. Interviewers watch for clean formulation of the random experiment, correct conditioning on which jar is chosen, and the insight that an intentionally skewed or asymmetric partition can strictly outperform intuitive even splits.

What it tests

This problem class is governed by the principle of maximizing expected probability through asymmetric partitioning, where you can create a scenario with a guaranteed win in one branch and a high-probability win in the other. When forced to make a random selection between two options, and you control the composition of each, the optimal strategy often involves isolating a single guaranteed-success outcome in one option (here, a jar with only a white marble), while concentrating the remaining resources to maximize the success rate in the other (the second jar). This leverages the fact that the overall probability is the average of the probabilities from each branch, so a certain win in one branch (probability 1) can be weighted against a less-than-certain but still favorable chance in the other. The pattern holds because splitting resources unevenly allows you to exploit the structure of the random choice, rather than diluting your chances by spreading resources evenly.