Tuesday, September 06, 2005

Help me get out of jail free?

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?

1. Not certain. The only solution i could think of is put only 1 white marble in bowl one. In bowl two put all 50 black marbles and the 49 remaining white marbles. You have a 50/50 chance of choosing bowl 1 which will give you a 100% chance to be free. The second bowl will give you a 49.49% chance to live. OK now... I need to get my statistics calculator.

2. Ok... with my previous guess.

WhiteBowl = 1
MixBowl = .494949

There is an even chance of picking either bowl so it is 1/2 the percentage of both bowls added together...

(.5*1)+(.5*.494949) = .5 + .2474745 = .7474745

So the best probability you could have is almost a 75% chance. Not bad. I would bet on vegas with those odds.

3. Karnov had the right answer! Congratulations! Mathematically, it's a matter of maximizing your chances where the formula is Z=.5*X + .5*Y, as Karnov has already pointed out.

4. What if you were to first put 50 black marbles in both bowls so that they are all on the bottom.. then you can place all of the white marbles on top of the black marbles. It wouldn't matter which bowl was chosen because you know that the white marbles are on top. ;)

5. Now that's thinking outside of the box... or should I say outside of the bowl. ;-)

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