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Life or Death? The Emperor's Proposition

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.


Puzzle ID:#15446
Fun:**** (3.15)
Difficulty:** (1.8)
Submitted By:pete**
Corrected By:Mogmatt16




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?

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Nov 19, 2003

in case anyone wants to know, this is about a 73.74% chance of living.
Nov 28, 2003

Fantastic teaser! Loved it! Sorry Mog the answer is just as described = 0.74 decimal repeater (74.74%) or precisely 74/99.
Nov 29, 2003

I would have placed all my black marbles in either or both bowls first, then placed the white marbles on top of the black marbles in both bowls. Thus when I pick a marble from the top, I would have very close to 100% probability of choosing a white marble.
Dec 01, 2003

Hey Parsifal,
I think that if you tried this, the King would see what you are up to and, being the sadistic SOB that he must be to set this whole thing up in the first place (), he would shuffle the balls around (after you are blindfolded) to put all of the black ones on top. The end result would be almost certain death.
Dec 13, 2003

a little too obvious for an adult, but maybe interesting for a kid.
Dec 16, 2003

Great teaser!
Dec 16, 2003

I like this one and realized that you have even more of a chance of picking the bowl with one white marble. When the emperor moves the bowl you will hear the one marble rolling around in the bowl all by itself and thus know which bowl to pick, the bowl with 99 marbles in it will not do that.
Dec 22, 2003

Good Teaser but kind of simple
Jan 19, 2004

that was sort of hard
Feb 13, 2004

Good teaser Made me think.
Apr 05, 2004

i totally agree with thongbutt. pink is such a pretty color. i love PINK ponys!
Aug 28, 2004

I was thinking exactly as parsifal was.
Oct 01, 2004

Which marble did the prisoner choose?
Nov 23, 2004

That was kinda fun but all u ppl who did the math have must have been bored.
Dec 16, 2004

one white in a bowl and the rest in the other.the chance to win would be nearly 75%
Dec 16, 2004

Hey infern0, you don't need to be bored to do most maths, just interested! Anyway, the maths in the this teaser took less than a minute, so you didn't even need to be that interested!
Dec 16, 2004

Why does it always seem as those who want you dead are so willing to let you live on some obscure technicality?
Dec 16, 2004

That is a puzzling question that even the sadistic person that I am can't even answer Mogmatt16 But this was a good early morning breakfast side-dish teaser.
Dec 16, 2004

Guys, I don't think the answer is correct. Your probability is still 50%.

Basically, you are asked to chose Marbles, not bowls and then marbles. If you list all possible outcomes eg: Bowl1,White; Bowl2,black; Bowl2,white; etc., you see that you still have 50 right choices and 50 wrong ones.
Dec 16, 2004

Sorry, Sreenadh, it's not 50%. The teaser says "You then can choose one bowl and remove ONE marble." When you pick a bowl, you have a 1/2 chance of getting the "white" bowl. If you get the "mixed" bowl, you have a 49/99 chance of getting a white marble. The total probability of living is the sum of all the other probabilities, namely (1/2 times 1) + (1/2 times 49/99). To show the work after that is 1/2*1 + 1/2*49/99 = 1/2 + 49/198 = 99/198 + 49/198 = 148/198 = 74/99, or the ~75% mentioned in other comments. A Monte Carlo program verifies this answer.
Dec 16, 2004

You can separate the blackand white marbles
Dec 16, 2004

As you blindly reach for the bowls, make sure to lightly bump both bowls. Simply select the lightest bowl and win.
(user deleted)
Dec 16, 2004

I disagree with the answer. Your probablilty would be even better putting two white marbles in one bowl, then you would have a 100% if picking from the light bowll and 50/98ths from the heavy bowll. This logic can be carried as far as one can distinguish the two bowls, to the limit of having 50 black+1 white in one bowl and 49 white in the other. Then one would have a 100% chance if given the light bowl (and you can discern that it is light) and a 50/51th chance of being correct with the heavy bowl. This gives approximately a 99% chance for survival and future revenge against the king.
Dec 16, 2004 do you figure? If you have 2 white marbles in one bowl, the the ratio of black to white in the other bowl is not 50/98. You only have 50 white to start with (unless the king is really evil and sneaky), so if you seperate 2 of them you would have a 48 out of 98 ratio in the other bowl. Your breakdown would be this: bowl 1 - 2 white, Bowl 2 - 48 white, 50 black. 2+48+50=100 marbles, 98 of which are in bowl 2.
Dec 16, 2004

Maybe he read the question wrong? lol..
Dec 16, 2004

this one was pretty interesting, and yet triky. I like it. Nice work.
Dec 16, 2004

Why couldn't you divided the black marbles in half and leave the white ones intact. That way no matter how many are in each bowl you will know to choose the whole marbles. It said you can divide the marbles anyway you want!!!
Dec 16, 2004

fastfingers i dont think thats what she meant but good answer, you really have to think outside the box for that one and great treaser pete!!
Dec 16, 2004

The riddle was easy for me, but thinking as a sinister king would think, which is what he would have to be to think this whole thing up....he would make sure that there wasn't any chance of you getting the white marble....think about it! Either way you divide the marbles if the king wants you dead..he wants you dead.
(user deleted)
Dec 16, 2004

no matter how you do it it's a 50 50 chance
Dec 17, 2004

Who cares about the math of this teaser , you have to admit that it is a good one!
Dec 17, 2004

but why can't you divide both black and white marbles equally in both bowls but start with all the black marbles at the bottom and the white marbles on top so you still have a 50/50 chance to pick a white marble? I liked the teaser because it had different possibilities.
Dec 17, 2004

yeah, we had one'o'these on an algebra test once. only it had bottles of poison. good teaser. but some'a yall had some pretty clever alternate answers.

by da way, mooses say quack just like piggies
Dec 20, 2004

very good teaser!!! but an EASY ONE!!
Jan 11, 2005

Classic! I love the probability ones.
Jan 20, 2005

good one!
Jan 29, 2005

its nice but i thought that it would be 25 black marbles and 25 white marbles in each bowl
May 07, 2005

Good teaser!
I didn't read the problem carefully enough and thought and thought each bowl had to have 50 marbles.
Jun 13, 2005

that was interesting, it's a good answer but i didn't even get close . . . oh well, good job
Dec 17, 2005

Love it! Hope you got white!
Dec 17, 2005

If the sun is shining, pick a cooler marble. If it isn't, put all the black ones in your pants for a few minutes before putting them back in the bowl.
Dec 17, 2005

I think if you just put one white marble in 1 bowl, 99 in the other bowl, u are guaranteed to win. Your answer is too complicated i think
Dec 17, 2005

Dec 17, 2005

Good teaser Mogmatt, i liked this one
Dec 17, 2005

This kind is too hard for me, I have some kind of strange resistance to percentages, but I appreciate it, Thanks!
Dec 17, 2005

Good teaser. I thought that he would have just put the white marbles on the top so he would know that the tops ones would equal life....
Dec 17, 2005

Like it!!! Couldn't work it out (no good at maths problems ) but I enjoyed the challenge!!!
Dec 17, 2005

Great teaser! But I think you have a 50/50 chance anyway you split them.
Dec 17, 2005

Heh heh... it doesn't look like that king was very bright. Ok, so he decides you should have a 50/50 chance to live. Wouldn't it have been simpler... and CHEAPER... to just flip a coin instead of going out to buy all those friggin' marbles? Also, the teaser didn't say whether you had to mix up whatever combination you chose... I'd just pour all the black marbles into each bowl first, and then pour all the white ones on top of all the black ones. And then just pick a TOP marble.
Dec 17, 2005

nice! i liked that one! good job.
Dec 17, 2005

It was ok, but kind of boring. Another way would have been to put a white marbel in one bowl and the rest in the other. Then just pick the marble out of the one that you know only has one marble, and you will for sure live.
Dec 17, 2005

Dec 17, 2005

if you put 25 black on bottom and 25 white on top you live
Dec 17, 2005

dat was preety cool..keep it up
Dec 17, 2005

Woo Hoo I got it!
(user deleted)
Dec 17, 2005

there is a better way... put 25 black in each bowl the put the white ones on top... no matter which bowl you choose, the white ones are on top
Dec 17, 2005

I think thatwas agood one! But whih bowl did eh choose from after all?
Dec 17, 2005

nice one
Dec 17, 2005

The weight of the bowl with 99 marbles would be 99 times greater than the other--at least I think?! Great fun. I love this sight!
Dec 17, 2005

simple and fun! Great job!
Dec 17, 2005

couldn't figure it out, but loved trying! great one
Dec 17, 2005

that one wasn't a very hard one
Dec 17, 2005

I liked this one--yeah it was easy, but those are the best ones.
Dec 17, 2005

Definitley a great guys say it was easy? Guess I'm just stupid....
Dec 17, 2005

thought it was fun and some of you....jimrcook....really need to relax and just enjoy the teaser.
Dec 17, 2005

Dec 17, 2005

At first I thought to put 25 white and 25 black in one bowl and then to put 25 whites and 25 blacks in the second bowl
Dec 17, 2005

It was ok. Alot of fun when you can figure it out. I did so it wasnt too hard.
Dec 17, 2005

Loved it and I got Makes a lot of SENSE!!!. A FAVORITE
Dec 17, 2005

I still think I like the idea of the 25 black marbles on the bottom of each bowl, and the 25 white ones on top...... (which I suggested 22 comments ago!!)... However, I think this was a teaser to get you to use mathematics to determine probability and outcome. Math was NEVER my strongest subject in school.
Dec 17, 2005

Dec 17, 2005

Probably shoulda had five cups of
coffee before trying this teaser!
Dec 17, 2005

A good teaser, I think there are a lot of different ways to solve this. I think everyone has their own idea, I was thinking the 25 bottom top thing, but the answer was good also. Anyway, fun teaser, I love these....
Dec 17, 2005

The only problem I can see with the 25 white on top and 25 black on bottom would be the size of the bowl. If the bowls are extremely large and the marbles are small then it wouldn't be possible to place them upon each other. I think we must rely on the math here.
Dec 17, 2005

Oh, by the way, I like this teaser even though I am not a fan of percentages or math.
Dec 17, 2005

you could have put 25 black marbles in each bowl(on the bottom), and 25 white on top, becouse they said mix the bowls around ,not the marbles.the whie ones would be right on top.
Dec 17, 2005

All the answers involving choosing the light bowl with one in it and taking the top marbles depend on things we don't know. Maybe the person had to just point at one of two spots, so would never get to feel if the bowl was heavy, hot, cool, light, etc. Then when they mixed the bowls around, if they did it with enough force all the top/bottom marbles would get mixed together, so you can't rely on that either.
Dec 17, 2005

loved this one! fun. way cool. What's with the obsession of marbles on top/bottom? Have you ever tried to "stack" marbles? Wouldn't the size of the bowl matter? To those who ask why those who want to kill give such chances-- great question---lol! Maybe the emperor wants the chooser to live.
Maybe emperor is bored & has nothing better to do with his time but think up absurd punishments-- just to see what happens. Ultimate power corrupts.
(user deleted)
Dec 18, 2005

umm... wouldnt the king/emporer (sp) just mix the bowl of marbles up... it doesn't say he cant... (if you put the white ones on the top and the black ones on the bottom)
Dec 19, 2005

Dec 19, 2005

Great teaser, these are the kind of teasers that are best because they make you think and after viewing the answer, you are either kicking yourself that you didn't get it or pleased with yourself that you did. Unfortunately, i do not have much experience with the latter. Once again, good teaser.
Dec 20, 2005

Ahh too much pressure!
Dec 22, 2005

This is a great teaser i loved it!
Jan 06, 2006

that was fun 'n easy I learn that in class already sorry but i still liked it
Feb 09, 2006

That was awesome! Going in my faves!
Feb 22, 2006

that was da bomb!
Mar 26, 2006

I thought that if you put one white in one, and in the other bowl put the whites one top you would get a better chance.
Apr 15, 2006

i love this !!!!!!!!!!!!!!!
i thought it was just great !
Dec 17, 2006

I got this right, but not for the same reason as in the official answer. I didn't think about the "choose one bowl" aspect as I walked to the bathroom, therefore I was thinking it would be easy to feel around, test bowl weight, or listen and tell the difference between 1 marble and 99.
Dec 17, 2006

What if you were to put all the black marbles on the bottom and then cover them with the white marbles, or is the emperor mixing the marbles around before letting you choose?
Dec 17, 2006

waste of time
Dec 17, 2006

Dec 17, 2006

You gotta love puttump or Imurry's self-valuation...As for the teaser, it work, simple and fun. Thank you. Good on you.
Dec 17, 2006

A very entertaining teaser. If you have to go through ALL of the math, ALL I can say is, you are not having fun... This kind of teaser has been around for ages, quit arguing for and against and have fun with it, or just totally hate it, and *Go away* Sheesh! BTW. a Totally Awesome Teaser. Have a marvelous day, Everyone! (or else enjoy the hatred of lmurray and
Dec 17, 2006

i didnt get it. thats kind of sad
Dec 17, 2006



Dec 17, 2006

OR you could put all of the whites in one bowl and all the blacks in one bowl...
Dec 17, 2006

Have to agree with Pizzazz. BTW I did get the official answer.
Dec 17, 2006

Dec 17, 2006

liked it! i didnt get it but once i read the answer i was like wow that makes sense!
lol i would love to see more
Dec 17, 2006

I had a hard time with this one because I kept thinking of being blindfolded and having 2 empty bowls. And I was supposed to sort the marbles which I could not see whether they were black or white. Hummmm. Needless to say I didn't get it. Thanks for submitting tho.
Dec 17, 2006

good teaser
Dec 17, 2006

That was a pretty awesome teaser but I've never been good with probability Even though I did get it wrong it was still pretty enjoyable though KEEP UP THE GOOD WORK!!!
Dec 17, 2006

Good teaser...basic logic, but still fun. I think the people who are getting confused are just misreading and misunderstanding the answer.
Dec 17, 2006

Dec 17, 2006

I think I'd have to go with 25 and 25
Dec 17, 2006

I was one of those that went with putting the black marbles on the bottom and the white on top. It's fun to read how other people wrapped their minds around this and came up with different answers. Just goes to show you that everything is not always "black and white"!
Dec 18, 2006

50/50 chance was the best I could do 25 of each color in each bowl and alot of praying
Dec 19, 2006

I actually played around with math a bit before the answer finally occurred to me, and my mathematical intuition told me that it was most certainly the right answer for the right reasons and I didn't even take the time to mathematically prove it to myself (after all, this is just a teaser, not a math test!)

I was not surprised to see suggestions regarding temperature of the marbles; I was pleasantly surprised to see the unique and clever suggestion about splitting the black marbles in half; I was appalled at how many people thought that stacking the white marbles on top would increase their chances. Maybe, just maybe, it would help to put the white marbles around the edges and the black ones in the middle. But I would imagine that the emperor is testing whether he should spare a smart guy's life, or put to death a smartass.

Anyways, it goes to show that you just never know when math might save your life. Yarrr.
Dec 19, 2006

Dec 20, 2006

Great teaser, but those who have some exposure to probability theory would find this quite an easy one.
Feb 19, 2007

i was thinking for of crapping and putting the crap on the black ones
Mar 21, 2007

Fun and easy!!!
(user deleted)
Mar 27, 2007

you also could put 50/50 in both bowls....putting the whites on top on both of them making the chance a 95% that you will live......
May 24, 2007

Even with those odds, i would still hate to be that guy!
(user deleted)
Aug 16, 2007

There is a much simpler way to guarentee that you live. Put one white marble into one bowl, and put the rest in the other bowl. When he asks you to pick one, feel the amount of marbles in both bowls and pick the marble from the bowl that only has one in it. Simple.
Sep 16, 2007

no, princerocker, you have to select only one bowl, you can't feel both of them
(user deleted)
Dec 02, 2007

if we place y marbles containing x white marbles in the first bowel then the chance of picking a white marble is:

1/2 * (x/y) + 1/2 * ((50-x)/(100-y))

such that 0 < x < 50 and 0 < y < 100

Find x,y satisfying the constraint above that will give the maximum value of this expression using Lagrange Multipliers.
Nov 18, 2008

While I enjoy some of the "outside the box" solutions (particularly the one involving splitting the black marbles in half!), let's assume that a random bowl is chosen, and a random marble is chosen from that bowl, with no opportunity for outside-the-box thinking. An easy way to prove that a distribution of 1 white marble in one bowl and the other 99 marbles in the other is optimal is as follows:

Maximizing your chances of picking a white marble is the same is MINIMIZING your chances or picking a black marble. The minimum chance of choosing any particular black marble is 1/99, which is achieved if and only if said black marble is in a bowl of 99 marbles. With a distribution of 1 white marble in one bowl and the other 99 marbles in the other, every individual black marble has this minimum chance of being drawn, so the overall chance of drawing one is also clearly minimized. QED.

As for the complaints of "why would an emperor give you an almost 75% chance of living?": Hey, this is a math puzzle, not an attempt at realism.
Nov 26, 2008

I thought about rearranging the marbles in tight horizontal rows and having a white or black on the right and left side of the bowl.
Mar 13, 2009

I loved this one... got it right, but my probability calculation is different than most. In my calculation, I found the total probability of drawing a white marble as 1/2 + 1/2(49/99) = (198 + 9 / (2 * 19 = 296/396 = 74.74747474... = 74.75%
Oct 10, 2009

Really good - thanks! I've tried a lot of probability questions and this was fun. I agree with your solution. 0.747474... is also correct.

I wonder if there ever were rulers, such as feature in these diversions, who, having nothing better for amusement, gave such straws for prisoners to clutch at!!
Dec 17, 2009

I suck at probability, but I instinctively figured out the correct answer in about 3 seconds. Fun problem though.
Dec 17, 2009

I lived. If you place 1 white marble in 1 bowl and the other 99 in the other bowl . T
Dec 17, 2009

Youre chance of survival is 100% . Place one white marble in 1 bowl the other 99 in the other bowl , then simply choose the lighter bowl. The teaser spefically says you get to choose the bowl .
Dec 17, 2009

With this kind of evil king, you get to choose the bowl by pointing to the left or right, not feeling the weight of the bowl. That said, you now have no chance to weigh the bowls. My answer was intuitive, and it was correct. Amazing for me. Great teaser, fun and doable even without all the math. Loved it. Thanks.
Dec 17, 2009

You would have slightly less than 75% if you put one white in one bowl and the rest in the other. However if you split them 50/50 in each you would have exactly 75% which would be greater.
Dec 17, 2009

"Mix the bowls" implied that the marbles could be remixed (at least to me). I would have understood the quiz had it said the emperor can "swap the two bowls".
Dec 17, 2009

If he was a prisoner, that means he had already lost his marbles!!!!
Dec 17, 2009

Dec 17, 2009

I didn't read the question properly, so my answer was just keep 1 white marble, distribute the rest randomly, and secretly pretend that you chose the white marble that you took
Jun 02, 2010

In one bowl put 1 white marble. In the other bowl put 50 black marbles at the bottom and 49 white marbles at the top. feel around at the top. If you feel amarble take it. If you don't go down to the bottom and pick that marble. 99% foolproof
(user deleted)
Apr 24, 2011

awesome teaser!
Dec 17, 2012

Am I the only one who has heard this one before? It is really an oldie, but then so am I and that is probably the reason I have heard it a long time ago! It has been around a while.
Dec 17, 2012

Right on Babe. This one was originated by Moses.
Dec 17, 2012

I have another answer:
Just put 25 marbles at the bottom of each bowl and cover them with 25 marbles each . You have a 99.9% chance of living.
Dec 17, 2012

No Casnie, that's the same answer that dozens before you have come up with. And it's just as wrong as the others because (a) you don't know the size of the bowl, and (b) you can't guarantee that the marbles won't rearrange themselves when the bowls are mixed.
Dec 18, 2012

In regard to the very first comment; it is not ~74%. It is 51% based on the answer.

Use a probability tree, and surely it will suggest the following:

Probability of picking:
X = Bowl 1
Y = Bowl 2
W = White marble
B = Black Marble

P(A)*P(W|A) + P(B)*P(W|B) = P(LIFE)

(1/2)*(49/49) + (1/2)*(1/51) = 51%

51% chance of picking the white marble, ladies and gentlemen.
Dec 18, 2012

Sorry guys, here' my comment, revised:

In regard to the very first comment; it is not ~74%. It is 51% based on the answer.

Use a probability tree, and surely it will suggest the following:

Probability of picking:
X = Bowl 1
Y = Bowl 2
W = White marble
B = Black Marble

P(X)*P(W|X) + P(Y)*P(W|Y) = P(LIFE)

"Place 1 white marble in one bowl, and place the rest of the marbles in the other bowl (49 whites, and 50 blacks)."

Translated mathematically yields:

(1/2)*(49/49) + (1/2)*(1/51) = 51%

51% chance of life, ladies and gentlemen.

Please DO message me if I am wrong and why.

Dec 18, 2012

Not sure where you're getting your probabilities from, Daryl, but the probability of choosing a white ball from the two bowls are, respectively, 1/1 and 49/99. Since there's a 50% chance of choosing either bowl, the probability of life is, as explained much, much earlier in the thread, (1/2)*1 + (1/2)*(49/99) = 74%
Dec 17, 2015

I would use one of these three methods.

Method #1
Place 25 black marbles in the bottom of each bowl. In each bowl place 25 white marbles on top of the black marbles. As mentioned above this will depend on the diameter of the bowl.

Method #2
Place one white marble in each bowl. Then spill the rest on the floor. As the guards are stumbling and falling on the marbles on the floor, make you escape. I strongly recommend grabbing a sword from one of the guards prior to departing the room. This works. Trust me.

Method #3
The magician's choice...
Randomly divide 50 of the black marbles and 49 of the white marbles between the two bowls. Palm one white marble. When the time comes to choose, hesitantly reach for one of the bowls then at the last moment switch to the other bowl. Reach in and pull out the palmed white marble. Do this with an over abundance of drama to make it appear that you are uncertain of the outcome. {NOTE:: For added drama place one white marble covered by all 50 black marbles into one bowl. In the other bowl place place "49" white marbles palming one of them in the process. Nervously explain to the Emperor that you are too scared to choose a bowl and ask him to choose the bowl.}
Dec 17, 2015

I'll accept the answer, but I personally don't feel the odds ever rise higher than 50%. With one white marble in the correct bowel, you have a 50% chance of picking the correct bowel. Of course if you pick the wrong bowel, you have less then a 50% chance as there are more black marbles than white ones in the "wrong" bowel. When observed as a whole you still have 50 white and 50 black, and therefore an equal chance of picking either bowel or either color marble.

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