T3B99

The story's kind of lame, but the problem's interesting:

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?

T3B98

The rightmost cup. 

The rightmost cup has a half chance of holding the coin, and the other cups have a quarter chance.

Pretend that Os represent cups, and Q represents the cup with the coin.

The game starts like this:

OOQ

Then your friend switches the rightmost cup with another, giving two possibilities, with equal chance:

OQO
QOO

Your friend then moves the cups again, but doesn't touch the rightmost cup. The only switch possible is with the leftmost cup and the middle cup. This gives two possibilities with equal chance:

QOO
OQO

Lastly, your friend switches the rightmost cup with another cup. If the first possibility shown above was true, there would be two possibilities, with equal chance:

OOQ
QOO

If the second possibility shown above (In the second switch) was true, there would be two possibilities with equal chance:

OOQ
OQO

^ I kinda got lost around here. (the purple text)
This means there are four possibilities altogether, with equal chance:

OOQ
QOO
OOQ
OQO

This means each possibility equals to a quarter chance, and because there are two possibilities with the rightmost cup having the coin, there is a half chance that the coin is there.

T3B97

I don't really get this.

You decide to play a game with your friend where your friend places a coin under one of three cups. Your friend would then switch the positions of two of the cups several times so that the coin under one of the cups moves with the cup it is under. You would then select the cup that you think the coin is under. If you won, you would receive the coin, but if you lost, you would have to pay.

As the game starts, you realise that you are really tired, and you don't focus very well on the moving of the cups. When your friend stops moving the cups and asks you where the coin is, you only remember a few things:

He put the coin in the rightmost cup at the start.

He switched two of the cups 3 times.

The first time he switched two of the cups, the rightmost one was switched with another.

The second time he switched two of the cups, the rightmost one was not touched.

The third and last time he switched two of the cups, the rightmost one was switched with another.

You don't want to end up paying your friend, so, using your head, you try to work out which cup is most likely to hold the coin, using the information you remember.

Which cup is most likely to hold the coin?

T3B96

I've heard this before, and the answer STILL blows my mind:

You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, "Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK... you will die."

How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?

T3B95

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The answer is... A word to the wise. - A word "to" the "y`s"

I can't really take out the answer in this image, so I've blocked it up as best as I can. Highlight the centre to find the answer!

I really like this one, it's pretty creative. :)

T3B94

Bananas. The letters you needed to cross out were: (S-I-X-L-E-T-T-E-R-S)

Classic. 

T3B93

Oh. Ha ha. I've kind of seen these before, but it never really registered in my brain until I looked at the answer.

Cross out six letters and you'll find a word that we should know. The word must be spelled out in order.

SBAIXNLETATNERSAS