### Brain Teasers

# The Tyrant King and the Marbles

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

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?

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?

### Answer

Suppose there are A red marbles in the first urn, B red marbles in the second urn, C red marbles in the third urn, and D red marbles in the fourth urn. Then your probability of drawing all red marbles is A*B*C*D/625, and your probability of getting at least one blue marble is (625-A*B*C*D)/625. The king will choose whichever option is more likely. To minimize the probability of a correct prediction, you want to get these values as close as possible to each other. Equivalently, you want A*B*C*D/625 to be as close as possible to 1/2, or A*B*C*D to be as close as possible to 312.5. Since A, B, C, and D can only take on the values of 0, 1, 2, 3, 4, and 5, the possibilities are limited. The closest possible value to 312.5 is 320=5*4*4*4; none of the closer values from 306 through 319 can be achieved because each one has a prime factor greater than 5. For example, 306=2*3*3*17, 307 is a prime, 308 = 2*2*7*11, and so on. Therefore, you should place 4 red marbles and 1 blue marble in three urns, and 5 red marbles in the remaining urn. King Fischer will predict that you will get all red marbles, and you'll have a 61/125=.488 probability of drawing a blue marble and surviving.In the end, the king takes some medicine to control his allergic reactions to much bleeding and executes you anyway. So much for all that mathematics and probability you just went through. What an anticlimax!

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