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## The Tyrant King and the Marbles

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #43634 Fun: (2.08) Difficulty: (2.76) Category: Probability Submitted By: mathgrant

You are currently being held prisoner by King Fischer, the tyrannical king of the faraway land of Fakefictio, for crimes involving a banana peel, a mechanical pencil, and the king's favorite Dragonforce album. Of course, as you know, you were not responsible for said crimes; these crimes never even took place. As a matter of fact, the king doesn't even like Dragonforce and is more of a fan of Journey and Kansas, but he is so tyrannical that he's arrested you anyway. The punishment for the crimes is a very graphic death that will probably involve much bleeding, and the date of your execution is today. The king could just kill you, but that wouldn't make for a very interesting puzzle at all, and he is allergic to much bleeding, so he decides to offer you a game to give you a chance to escape with your life. Win, and you will go free. Lose, and you will not only not go free, but you'll probably die, too.

The king presents to you four urns and a bowl filled with red and blue marbles. As two guards pin you to the dungeon wall by the arms, a third guard will place 5 marbles in each urn. You get to choose the distribution of red and blue marbles in each urn. For example, you could have the guard place 3 red and 2 blue marbles in one urn, and 1 red and 4 blue in another urn. You can also place 5 red or 5 blue in a single urn. King Fischer's watchful eye will observe the process, so he will know your distribution.

After all of the urns have 5 marbles each, the guard will shake the urns, and the king will predict whether you will draw all red marbles or at least one blue marble. The guards will then cover your eyes with a blindfold and place your hand in each urn; you will be required to draw exactly one marble from each urn. If the king's prediction proves correct, you will be executed. However, if the prediction is incorrect (that is, you draw a blue marble when the king predicts you'll draw all red marbles, or vice versa), then you will go free.

Note that if you put 5 red marbles in every urn, then the king will know for certain that you'll draw all red marbles, and use this knowledge to your disadvantage to get you killed. Contrariwise, if you put 5 blue marbles in one urn, then you'll be forced to draw at least one blue marble, and the king will predict this, too. In order to have hope of survival, you'll need to distribute the marbles in a fashion that makes the outcome less certain.

How should you distribute the marbles to maximize your chances of survival, and what are your chances of survival under this strategy?

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