You are a prisoner sentenced to death

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  • #1
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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 and marbles up. You can then choose one bowl and remove one marble. If the marble is white, you live, but if the marble is black...you die.

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

I'm not sure if it's been posted before but I liked it...
 

Answers and Replies

  • #2
664
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Then I will blindfold you and mix the bowls and marbles up.
I assume this is meant to simply mix the bowls up, and not the marbles? That is, the marbles will not be moved from one bowl to the other-- merely have their positions in their already-assigned-bowls moved around so that (for example) one at the top might wind up somewhere in the middle, and so forth.

I get a 74.747474... percent chance of living...

DaveE
 
  • #3
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I assume this is meant to simply mix the bowls up, and not the marbles? That is, the marbles will not be moved from one bowl to the other-- merely have their positions in their already-assigned-bowls moved around so that (for example) one at the top might wind up somewhere in the middle, and so forth.

I get a 74.747474... percent chance of living...

DaveE
yeah.

how did you arrange the marbles. (its right by the way)
 
  • #4
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I get 98% minimum.
Put one white marble in one of the bowls and leave the rest in the other.
 
  • #5
381
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I get 98% minimum.
Put one white marble in one of the bowls and leave the rest in the other.
I'm pretty sure that would give you 74.74747474% chance of survival. It's 50/50 (so equal) that you choose one or the other bowl. If you pick one bowl you have 100% chance of survival. If you choose the other bowl you have 49/99 so basiaclly 49.494949% Since both have equal probability of occuring you add them and divide by total amount of percentage (so 200%)

100%+49.4949...%=149.4949...%

149.4949.../200= 74.7474...% chance of survival. It's completely possible however my math is totally wrong. How did you end up with 98%?
 
  • #6
Matterwave
Science Advisor
Gold Member
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I get 100%

Put one white marble in one, all the rest in the other. Blindfolded, pick the bowl that only has 1 marble in it... <_<

If you're not allowed to feel around, then I'm not sure atm.
 
  • #7
664
3


It's completely possible however my math is totally wrong.
The math is correct, although I don't think it's quite how one would typically express the probability (as in, the way probability is taught-- I could be wrong, I'm not sure. It's been a while since my last class on the subject). The 4 outcomes would be:

A) Pick a white marble from bowl 1
B) Pick a black marble from bowl 1
C) Pick a white marble from bowl 2
D) Pick a black markble from bolw 2

So, to start with, there's a 1/2 chance of picking bowl 1. Then, assuming that bowl 1 has a single white marble, there's a 1/1 chance of picking a white marble, and a 0/1 chance of picking a black marble.

Next, there's a 1/2 chance of picking bowl 2. And a 49/99 chance of picking a white marble from bowl 2, with a 50/99 chance of picking a black marble from bowl 2.

Hence:

A) 1/2 * 1/1 = 0.5
B) 1/2 * 0/1 = 0.0
C) 1/2 * 49/99 ~ 0.24747474...
D) 1/2 * 50/99 ~ 0.25252525...

The overall chance of picking a white marble
= A+C
= 1/2 * 1/1 + 1/2 * 49/99
~ 0.5 + 0.24747474...
~ 0.74747474...

Or, the alternate way (often easier in probability problems) is to realize that there's only 1 viable way of picking a black marble, which is to pick it from bowl 2. Hence, you can just state:

Overall chance of picking a white marble
= 1 - chance of picking a black marble
= 1 - 1/2 * 50/99
~ 1 - 0.25252525...
~ 0.74747474...

DaveE
 
  • #8
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Ah thanks daveE, I only did it that way though because I already knew that both bowls were on equal footing. Your way is correct though.
 
  • #9
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I'm pretty sure that would give you 74.74747474% chance of survival. It's 50/50 (so equal) that you choose one or the other bowl. If you pick one bowl you have 100% chance of survival. If you choose the other bowl you have 49/99 so basiaclly 49.494949% Since both have equal probability of occuring you add them and divide by total amount of percentage (so 200%)

100%+49.4949...%=149.4949...%

149.4949.../200= 74.7474...% chance of survival. It's completely possible however my math is totally wrong. How did you end up with 98%?
Why would you average the probabilities together?

But, yes, daveE is right. I just noticed that I did the math wrong -_- it should be 74.74%
 
  • #10
381
0


Why would you average the probabilities together?
Because they are on even footing (since it's 50/50)? It's definitely not wrong. Why would I want to waste my time figuring out what the percentage value out of 100% is and then figure the percentage of 50% that represents and then add them when I could just do it all at once?
 
  • #11
180
1


I get 100%

Put one white marble in one, all the rest in the other. Blindfolded, pick the bowl that only has 1 marble in it... <_<

If you're not allowed to feel around, then I'm not sure atm.
I'm with you on this one. There's nothing in the instructions which say they have to be divided equally, or that you're not allowed to reach for the bowls and pick the marble out yourself.
 

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