"let's make a deal" Marbles

[BMAD]

you are a contestant on a game show competing against another person. There are four jars each containing 20 marbles. At the start, one jar has all blue marbles, another has only green marbles, a third has white marbles, and the last has black marbles. You see which jar has which at the beginning (the other contestant does not).

You may redistribute the marbles however you like except three conditions must persist:

• every jar must contain at least 1 marble,
• every marble must be in a jar,
• no jar may contain more than 50 marbles.

Once the redistribution occurs, you and the other person will be blindfolded and jars shuffled and randomly placed on the table. They will pick one jar first then you will pick one from the remaining three. The marbles in each jar are then counted and scored according to their values:

• blue = 1
• green = 3
• white = 5
• black = 10

Two questions:

1. What would be the best configuration of marbles that would most help assist you in winning?
2. What is the probability you would win this game?

Edited May 17, 2014 by BMAD

[bubbled]

Let's add one more caveat then....

After the first person choises, you have the option to redistribute the marbles again.

First, I would distribute all the marbles equally among the 4 jars. But, then I would take 1 black marble out of jars 2, 3 and 4 and place them in jar 1 before commencing the game.

I'm assuming that during the redistribution phase, I will know whether my opponent chose jar 1, 2, 3 or 4.

If my opponent chose jar 1, I can easily redistribute the remaining marbles, such that I will have 2/3 chance to win. Basically, I will load up on two jars, leaving the other one almost empty.

If my opponent chooses any of jars 2, 3 or 4, I will simply take two black marbles out of jar 1, and place one in each of the other two jars. Now all three jars have larger value than the jar my opponent chose. I am guaranteed the win.

My overall odds of winning are (1/4)*2/3 + (3/4)*1 = 11/12 or 91.67%

[bonanova]

Your opponent does not see which jars contain which colors at first. Seems not to matter since they are later scrambled.

Your opponent picks first. Seems not to matter since the jars were randomized.

But other things might matter:

1. Are the jars are transparent?
   
   Does your opponent know:
   
2. That there are four jars?
3. That there are four colors?
4. That there are 20 marbles of each color?
5. What the colors are worth?
6. What the "redistributed jars looks like, generally (assuming they're transparent) before being blindfolded?

Or maybe none of that matters. All your opponent does is blindly grab a jar. That action does not use knowledge.

You assign point values to each jar (with restrictions) and then pick one that's higher than the one your opponent blindly picked.

Interesting, because the contest seems to have symmetry, and there doesn't seem to be a strategy to break it.

[BMAD]

Your opponent does not see which jars contain which colors at first. Seems not to matter since they are later scrambled.

Your opponent picks first. Seems not to matter since the jars were randomized.

But other things might matter:

1. Are the jars are transparent?

   Does your opponent know:

2. That there are four jars?
3. That there are four colors?
4. That there are 20 marbles of each color?
5. What the colors are worth?
6. What the "redistributed jars looks like, generally (assuming they're transparent) before being blindfolded?

Or maybe none of that matters. All your opponent does is blindly grab a jar. That action does not use knowledge.

You assign point values to each jar (with restrictions) and then pick one that's higher than the one your opponent blindly picked.

Interesting, because the contest seems to have symmetry, and there doesn't seem to be a strategy to break it.

Each jar feels identical (since you and your contestant are blindfolded the look shouldn't matter)

No matter how many marbles they weigh an imperceptible difference

The other person only knows the amount of jars and the starting count and colors in each jar and of course knows the value of each marble.

I did leave this part out, you are only blindfolded after the first person picks their jar and then the jars are shuffled again while you are blindfolded.

The jars are not transparent

[bonanova]

OK I'll have to think about it some more.

It seems impossible to implement a strategy.

You're blindfolded and grab one of three jars your opponent didn't pick.

Now, to state that problem in a way that will allow a solution ....

[BMAD]

Let's add one more caveat then....

After the first person choises, you have the option to redistribute the marbles again.

[dgreening]

I've dabbled with various strategies:

• Stragegy 1:
  • initial distribution: all jars have equal points [95] and
  • 
    
  • second distribution: 2 jars have 142 points, 1 jar has 1
    point
• 
  
• Strategy 2:
  • initial distribution: 1 jar has a relatively large number of points
    [say 140], 3 jars have equal points [80] and
  • 
    
  • second distribution: 3 jars have equal points

They all seem to have the same expected pay off [95 points to each player] if
the game is played many times

[dgreening]

One more question:

Does it matter how many points you win by?? [say you were paid $1 per point]; or

Is winning by 1 point as valuable as winning by 20 points?

[BMAD]

Winning is winning

[dgreening]

Winning is winning

In that case, I think bubbled has the optimum strategy.

[bonanova]

you are a contestant on a game show competing against another person. There are four jars each containing 20 marbles. At the start, one jar has all blue marbles, another has only green marbles, a third has white marbles, and the last has black marbles. You see which jar has which at the beginning (the other contestant does not).

You may redistribute the marbles however you like except three conditions must persist:

• every jar must contain at least 1 marble,
• every marble must be in a jar,
• no jar may contain more than 50 marbles.

Once the redistribution occurs, you and the other person will be blindfolded and jars shuffled and randomly placed on the table. They will pick one jar first then you will pick one from the remaining three. The marbles in each jar are then counted and scored according to their values:

• blue = 1
• green = 3
• white = 5
• black = 10

Two questions:

1. What would be the best configuration of marbles that would most help assist you in winning?
2. What is the probability you would win this game?

The solution contains the assumption:

I'm assuming that during the redistribution phase, I will know whether my opponent chose jar 1, 2, 3 or 4.

I am trying to reconcile that with the red portion of the OP

[bubbled]

you are a contestant on a game show competing against another person. There are four jars each containing 20 marbles. At the start, one jar has all blue marbles, another has only green marbles, a third has white marbles, and the last has black marbles. You see which jar has which at the beginning (the other contestant does not).

You may redistribute the marbles however you like except three conditions must persist:

• every jar must contain at least 1 marble,
• every marble must be in a jar,
• no jar may contain more than 50 marbles.

Once the redistribution occurs, you and the other person will be blindfolded and jars shuffled and randomly placed on the table. They will pick one jar first then you will pick one from the remaining three. The marbles in each jar are then counted and scored according to their values:

• blue = 1
• green = 3
• white = 5
• black = 10

Two questions:

1. What would be the best configuration of marbles that would most help assist you in winning?
2. What is the probability you would win this game?

The solution contains the assumption:

I'm assuming that during the redistribution phase, I will know whether my opponent chose jar 1, 2, 3 or 4.

I am trying to reconcile that with the red portion of the OP

I was with you as to there being no way to gain an advantage until BMAD added this to the terms of the problem:

After the first person chooses, you have the option to redistribute the marbles again.

Given, that you get to redistribute the marbles, it is reasonable to assume you can tell which jar was chosen and then act accordingly.

[bonanova]

you are a contestant on a game show competing against another person. There are four jars each containing 20 marbles. At the start, one jar has all blue marbles, another has only green marbles, a third has white marbles, and the last has black marbles. You see which jar has which at the beginning (the other contestant does not).

You may redistribute the marbles however you like except three conditions must persist:

• every jar must contain at least 1 marble,
• every marble must be in a jar,
• no jar may contain more than 50 marbles.

Once the redistribution occurs, you and the other person will be blindfolded and jars shuffled and randomly placed on the table. They will pick one jar first then you will pick one from the remaining three. The marbles in each jar are then counted and scored according to their values:

• blue = 1
• green = 3
• white = 5
• black = 10

Two questions:

1. What would be the best configuration of marbles that would most help assist you in winning?
2. What is the probability you would win this game?

The solution contains the assumption:

I'm assuming that during the redistribution phase, I will know whether my opponent chose jar 1, 2, 3 or 4.

I am trying to reconcile that with the red portion of the OP

I was with you as to there being no way to gain an advantage until BMAD added this to the terms of the problem:

After the first person chooses, you have the option to redistribute the marbles again.

Given, that you get to redistribute the marbles, it is reasonable to assume you can tell which jar was chosen and then act accordingly.

I did leave this part out, you are only blindfolded after the first person picks their jar and then the jars are shuffled again while you are blindfolded.

The jars are not transparent

With this order of events:

1. I fill the jars the way I want them
2. Because the jars are not transparent my opponent has no clue about their contents
3. My opponent picks one
4. I know which one he picked, either because I was watching

   or because I now have access to all the remaining jars and marbles.

5. I re-fill the jars the way I want them.
6. I am blindfolded
7. The jars are shuffled
8. I pick one, while blindfolded

there is a strategy.

[bubbled]

you are a contestant on a game show competing against another person. There are four jars each containing 20 marbles. At the start, one jar has all blue marbles, another has only green marbles, a third has white marbles, and the last has black marbles. You see which jar has which at the beginning (the other contestant does not).

You may redistribute the marbles however you like except three conditions must persist:

• every jar must contain at least 1 marble,
• every marble must be in a jar,
• no jar may contain more than 50 marbles.

Once the redistribution occurs, you and the other person will be blindfolded and jars shuffled and randomly placed on the table. They will pick one jar first then you will pick one from the remaining three. The marbles in each jar are then counted and scored according to their values:

• blue = 1
• green = 3
• white = 5
• black = 10

Two questions:

1. What would be the best configuration of marbles that would most help assist you in winning?
2. What is the probability you would win this game?

The solution contains the assumption:

I'm assuming that during the redistribution phase, I will know whether my opponent chose jar 1, 2, 3 or 4.

I am trying to reconcile that with the red portion of the OP

I was with you as to there being no way to gain an advantage until BMAD added this to the terms of the problem:

After the first person chooses, you have the option to redistribute the marbles again.

Given, that you get to redistribute the marbles, it is reasonable to assume you can tell which jar was chosen and then act accordingly.

I did leave this part out, you are only blindfolded after the first person picks their jar and then the jars are shuffled again while you are blindfolded.

The jars are not transparent

With this order of events:

1. I fill the jars the way I want them
2. Because the jars are not transparent my opponent has no clue about their contents
3. My opponent picks one
4. I know which one he picked, either because I was watching

   or because I now have access to all the remaining jars and marbles.

5. I re-fill the jars the way I want them.
6. I am blindfolded
7. The jars are shuffled
8. I pick one, while blindfolded

there is a strategy.

Yes. Your breakdown is exactly how I interpreted the problem after the clarifications. Do you disagree with my strategy?

[bonanova]

No.