T3B100

I don't get it.

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!

And this is a sad conclusion. :(

*Also, can you be allergic to bleeding? Wouldn't you have to be allergic to blood? I don't think you can be allergic to an action...