You are a prisoner in a strange land, and you have been sentenced to death. However, the ruler of the land is willing to give you a chance to escape. He will place two jars in front of you, one containing 50 black marbles and the other containing 50 white marbles. You will choose one jar at random, and then you must draw marbles from that jar until you have drawn 10 marbles. If you manage to draw all 10 marbles of the same color, you will be set free. If not, you will be put to death.
However, the ruler of the land is not entirely fair. He has secretly removed one of the marbles from one of the jars, making it impossible to draw all 10 marbles of the same color from that jar. You do not know which jar has the missing marble.
How can you maximize your chances of choosing the jar with the missing marble and therefore increase your chances of survival?
To maximize your chances of survival, you should choose a jar and then draw a single marble from that jar. If the marble you draw is white, choose the jar that originally contained the black marbles. If the marble you draw is black, choose the jar that originally contained the white marbles.
This strategy works because if the jar you initially choose has the missing marble, you will not be able to draw 10 marbles of the same color no matter which jar you choose. In this case, you will not survive regardless of which jar you choose.
However, if the jar you initially choose does not have the missing marble, then the other jar must have it. By drawing a single marble and using that information to choose the other jar, you have increased your chances of selecting the jar with the missing marble. In fact, your chances of selecting the jar with the missing marble are now 49/99 or approximately 49.5%.
After you choose the second jar, continue drawing marbles until you have either drawn 10 marbles of the same color and are set free or have failed to do so and are put to death.