Only people with an IQ of 115 or higher can get this right

[dyna]

The difference between this problem and the Bertrand problem is described below and why it is creating confusion.

The Bertrand problem is without a doubt 2/3. The OP's question would follow the logic of the Bertrand problem if it stated the question first, for example:

What is the probably that the next ball you take from the same box will also be gold if there are 3 boxes......

Which would align with the way the problem was stated on wikipedia.

The 'paradox' is in the probability, after choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, of the next coin drawn from the same box also being a gold coin.

However,

He asked the probability question at the end and it adds some ambiguity to when the probability starts. The 50% crowd sees the probability start when you have a gold ball. That means you only have 2 boxes to choose from and therefore it is 50%. The rest of the information about the 3 boxes is irrelevent.

I think someone made the point earlier that the answer could be both 1/2 and 2/3 depending on how you read the question.

If you create another thread and restate the question at the beginning forcing to consider all the variables including the 3 boxes, you're probably going to get agreement that it is 2/3.

 

[Red Squirrel]

What if the boxes are on a plane, which is on a treadmill. When the plane tries to take off, will the balls inside the boxes roll to the walls of the boxes or will everything all move in unison with the rest of the plane? Will this movement change the odds of which balls you end up grabbing?

 

[William Gaatjes]

What is interesting is why does the probability stay the same before and after the event that you have picked a gold ball and have to pick from the same box.
It is interesting to think about that you have as much chance as getting G1, and then G2, as G2 and G1, and G3 and S1, and S1 and G3.
I wonder if there is a physics experiment that also intuitively makes people guess wrong.

 

[DigDog]

You are still more likely to have drawn the gold ball from the box with 2 gold balls which makes it 2/3.

We're trying to explain to you that only states with G1S exist but whatever.

 

[William Gaatjes]

Makes me think : It is when separate items are grouped together and seen as one while still separate at the same time that suddenly the probability changes.

 

[richaron]

...However,

He asked the probability question at the end and it adds some ambiguity to when the probability starts. The 50% crowd sees the probability start when you have a gold ball. That means you only have 2 boxes to choose from and therefore it is 50%. The rest of the information about the 3 boxes is irrelevent.

I think someone made the point earlier that the answer could be both 1/2 and 2/3 depending on how you read the question.

If you create another thread and restate the question at the beginning forcing to consider all the variables including the 3 boxes, you're probably going to get agreement that it is 2/3.

*bold added.

I've tried to explain it, but maybe it will work if I just say "nope". You are wrong. There is no ambiguity. You have gotten confused because you don't understand the question or you have gotten confused because you have listened to people who try to people who are trying to create ambiguities.

The question in the OP is "...What is the probability that the next ball you take from the same box will also be gold?"

The question is about "the next ball". As has been stated ad nauseam there are 3 possible "next ball"s out of the 4 total balls in the two boxes which contain at least one gold ball. If, as you state, "the 50% crowd sees the probability start when you have a gold ball" then they only need to read my previous sentence an learn where they went wrong. Two boxes does not equal 50% when there are 3 possibilities. The possibilities are only narrowed down to 3 "when you have a gold ball".

In no way, shape, or form is the question ambiguous. The question however can be interpreted the wrong way...

 

[DigDog]

Bertrand: so, you have these boxes.
You: ok.
Bertrand: and one has a silver ball in it.
You: ok.
Bertrand: and you draw a ball.
You: ok i draw the silver ball.
Bertrand: no that cannot happen.
You: i am not allowed to draw the silver ball?
Bertrand: no, this experiment exists in a reality where you are not allowed to draw silver.
You: ok.
Bertrand: now calculate the possibilities.
You: this much.
Bertrand: Aha! You fool, you didnt consider the possibility of drawing silver!!
You: ...

 

[DigDog]

The question is about "the next ball". As has been stated ad nauseam there are 3 possible "next ball"s out of the 4 total balls in the two boxes which contain at least....

The silver box can exist in one state only; the gold box can exist in ONE of two states which are half as likely each as the single state of the gold box.

Im not asking if you agree, only if you understand what i wrote.

 

[DigDog]

You are still more likely to have drawn the gold ball from the box with 2 gold balls which makes it 2/3.

Only in instances where you could draw silver first.

 

[richaron]

The silver box can exist in one state only; the gold box can exist in ONE of two states which are half as likely each as the single state of the gold box.

Im not asking if you agree, only if you understand what i wrote.

Nope, sorry I don't. At what stage are you talking about?

At the point of the question I see the gold box existing in a single state which is twice as likely as the mixed box. I don't imagine the gold box ever existing in multiple states, but there are two possibilities of picking a gold from the gold box and each probability is the same as any other option... .

Edit: I mean.. I have done tertiary level stats and physics so I sort of get the mindset you are presenting (it reads like quantum level stuff), I just don't think it's particularly useful or relevant to the problem at hand. And I think it's easy to confuse oneself getting too abstract. Which is why I have consistently been falling back to the first principles of statistics.

 

[momeNt]

Why is this giving people a hard time?

You don't count the remaining outcomes, you sum the probability of the outcomes happening from the beginning. Like holy shit people. This is ATOT, we are better than this.

 

[richaron]

Why is this giving people a hard time?

You don't count the remaining outcomes, you sum the probability of the outcomes happening from the beginning. Like holy shit people. This is ATOT, we are better than this.

*bold added

This is also worded... poorly?

Seems ATOT is exactly this

 

[Humpy]

...next ball ... same box ... next balls... two boxes

I see one box.

 

[Humpy]

You don't count the remaining outcomes, you sum the probability of the outcomes happening from the beginning.

How exactly was the "beginning" determined? What if you were explicitly told the beginning was somewhere else?

 

[loafbred]

Only in instances where you could draw silver first.

It seems to me that you're stating that silver was not available in the first place. It was available. Picking a box, which constitutes the first event, was not explicitly excluded from the problem. That event, alone, changes the probability from 1/2 to 2/3. What you're trying to do is comparable to stating "a gold ball was purposely chosen to simplify the problem".

 

[Tweak155]

First, I wrote "their right answer" not "a right answer". And I don't agree that I can't have an answer.

Second, I don't know how many answers there are, if any. The right answer may exist, maybe more than one, maybe none. I'm okay with that.

I'm more interested in questions than answers anyways, which is why I like threads like these.

Ok, it may have helped to put the word right in quotes or indicated that you question that there is a right answer in that reply. All good.

As long as you're consistent in your message

 

[Schfifty Five]

It's 2/3.

The box with 2 gold balls has 2 outcomes. You could have picked Gold #1 and then Gold #2, or Gold #2 then Gold #1, different outcomes. Then there is the 3rd outcome where you pick the gold ball from the box that has a silver.

So 3 outcomes, 2 will result in a 2nd gold ball. Thus 2/3 that the 2nd ball will also be gold.

 

[loafbred]

It's 2/3.

The box with 2 gold balls has 2 outcomes. You could have picked Gold #1 and then Gold #2, or Gold #2 then Gold #1, different outcomes.

I've seen this stated several times by two or three different people, and it is patently irrational. It makes no difference which ball you might pick from the G+G box, because they are identical. You can number them, name them, and imagine any number of hidden differences between them, but that cannot alter their influence on the outcome individually. Their similarity is a factor, but neither their position in the box, nor their order of being picked from the box, are relevant. It is only important that they are the same.

 

[JujuFish]

2/3

The plane takes off.

.9... = 1

 

[Hitman928]

I've seen this stated several times by two or three different people, and it is patently irrational. It makes no difference which ball you might pick from the G+G box, because they are identical. You can number them, name them, and imagine any number of hidden differences between them, but that cannot alter their influence on the outcome individually. Their similarity is a factor, but neither their position in the box, nor their order of being picked from the box, are relevant. It is only important that they are the same.

In reality, true. It just helps some people visualize the fact that there are two distinct objects in the box that add to the probability rather than 1. Whether you label them or not, it doesn't make a difference, but it helps the brain "see" why the answer works out the way it does.

 

[loafbred]

In reality, true. It just helps some people visualize the fact that there are two distinct objects in the box that add to the probability rather than 1. Whether you label them or not, it doesn't make a difference, but it helps the brain "see" why the answer works out the way it does.

I appreciate their reason for organizing them in a diagram, but him stating that pulling them from different positions, in the same box, will create two different outcomes, prompted me to argue.

 

[BudAshes]

 

[DigDog]

It seems to me that you're stating that silver was not available in the first place. It was available. Picking a box, which constitutes the first event, was not explicitly excluded from the problem. That event, alone, changes the probability from 1/2 to 2/3. What you're trying to do is comparable to stating "a gold ball was purposely chosen to simplify the problem".

Thats exactly what happened. If you state "you pick either box, and draw gold" that is the state you are describing.

 

[Darwin333]

Sigh. This has gotten to the point of the plane on a treadmill argument. We have convinced all of those that had open minds and have seen the proof, the rest will stick to their guns no matter how much evidence, proof or professional opinion is given to them. So let them think they are smarter than everyone else, including people with PHDs in the subject...

 

[Cozarkian]

He asked the probability question at the end and it adds some ambiguity to when the probability starts. The 50% crowd sees the probability start when you have a gold ball.

But that's precisely why it is 2/3. If the probability started before you had the gold ball the odds of choosing the double gold box would be 1/3. It isn't until after you discover the ball you picked is gold that you can recalculate the odds with new information.

That means you only have 2 boxes to choose from and therefore it is 50%.

Are you sure? There are two boxes, gold/gold and gold/silver. Before drawing, your prospective odds of picking either box is 50/50, and your odds of picking any one ball are 25%. If you pick a ball and see it is gold, you now know there is a 2/3 chance you have the gold/gold box (alternatively, if you pick a silver, you know there is a 100% chance you have the gold/silver box).

This is because there are two ways you could have drawn a gold ball if you had box 1 and only 1 way you could have drawn a gold ball if you had box 2. Remember, the odds were Box 1: 25% + 25% = 50% verse Box 2: 25% + 25% = 50%. Since you didn't pick silver, you can reduce the odds of silver to 0%, which leaves Box 1: 25% + 25% = 50%, Box 2: 25% + 0% = 25%, which is a 2:1 ratio between the remaining choices, yielding a 2/3 chance that you have Box 1 and 1:3 chance you have Box 2.

Still don't see it? Add balls. There are two boxes, one has 100 gold balls the other has 1 gold ball and 99 silver balls. You draw a ball randomly from a random box and it is gold. True, there was a 50% chance you would select the box with 99 silvers, but after drawing a gold ball, would you still think there are equal odds you have a box with 99 silvers?