Marbles Probability Puzzle

Marbles Probability Puzzle

You are a prisoner sentenced to death.
The Emperor offers you a last chance to live by playing a simple game.
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 saving your life that is choosing a WHITE marble?

Solution

Say we put all the White marbles into JAR A and all the Black ones into JAR B. then our chances for picking a red one are:

• 1/2 chance we pick JAR A * 50/50 chance we pick a White marble
• 1/2 chance we pick JAR B * 0/50 chance we pick a White marble

You would try different combinations, such as 25 of each colored marble in a jar or putting all white  marbles in one jar and all the black in the other. You would still end up with a chance of 50%. What if you put a single white marble in one jar and the rest of the marbles in the other jar? This way, you are guaranteed at least a 50% chance of getting a white marble (since one marble picked at random, doesn’t leave any room for choice). Now that you have 49 white marbles left in the other jar, you have a nearly even chance of picking a white marble (49 out of 99).

So the maximum probability will be :

• Jar A : (1/2)*1 = 1/2 (selecting the jar A = 1/2, White marble from jar A = 1/1)
• Jar B : (1/2)*(49/99) = 0 (selecting the jar B = 1/2, White marble from jar B = 49/99)

Total probability = 74/99 (~3/4) 