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Tech giant Amazon reportedly poses a series of logical and mathematical puzzles during the interview process in order to select the best candidates, including the following lottery scenario. Take your best shot, then scroll down for the solution.
You have 100 red balls, 100 blue balls, and 2 urns. You distribute all of the balls between both urns, placing at least 1 ball in each, you cannot place them all in the same urn. You then pick one of the urns at random, and select a ball from it. You win $ 100 if you pick a red ball. With that in mind, what is the best strategy for distributing the balls, and what is your best winning percentage?
Presh Talwalkar lays out the optimal strategy in his YouTube video.
To be in with the best chance of winning, Talwalker places 1 red ball in one of the urns, and 99 red balls and 100 blue balls in the other. That gives you a 100% probability of picking a red ball from the first urn. For the second urn, there is a 99/199 (49.7%) probability.
As you are choosing an urn at random, you are equally likely to get one of these probabilities, so your winning percentage is the average of these two cases, which is 149/199, or 74.87%.
“This intuitively seems like the best answer,” says Talwalkar, “but how do we know that?” He goes on to prove the optimality of this strategy by considering other approaches. For example, distributing the balls evenly between the urns gives you a 50% chance in each case, and a 50% chance overall of picking red. In another instance, putting more red balls in one urn than another creates a far lower probability when picking from the urn with predominantly blue balls.