There are 3 pancakes in a hat: one is golden on both sides, one is brown on both sides, and one is golden on one side and brown on the other. You reach in, pull out a pancake, plop it on the table, and the side you can see is golden. What's the probability that the other side is also golden?
This market resolves to that probability.
(We'll determine this by consensus in the comments, which I'm confident we will achieve this time because this problem is completely straightforward. We can simulate it to convince people if needed! I won't trade in this market so I can be a disinterested adjudicator juuuuust in case consensus proves tricky.)
1,000
3.00There are 6 equiprobable outcomes. The revealed side is:
First pancake golden
First pancake golden, but the other side
Second pancake brown side 1
Second pancake brown side 2
Third pancake golden side
Third pancake brown side
By gaining knowledge that the revealed side is golden we can truncate the unsuitable cases 3 4 6.
We get:
First pancake golden side
First pancake golden other side
Third pancake golden side.
Out of these 3 outcomes which are still equiprobable the first 2 will have the other side golden. The 100% probability is redistributed to those 3 outcomes keeping their probability proportions.
Answer = 2/3.
But I do not agree that this problem is related to any other one discussed within the month.
In my recent "2 dollar" example (in answer to which you mentioned this market) those dollars are not equiprobable, their probabilities are tied to a coin flip, they have to match the probabilities of the coin. The existance of two dollars on your property is 50%. The existance of a single dollar is 50%. When you find a dollar (you cannot tell which dollar it is) no outcome can be truncated, because every universe contains at least a dollar. If nothing is truncated, then probabilities are fully distributed futher downstream: it is 50% that is Heads universe and you found the dollar, it is 25% that it is Tails universe and you found the first out of two signed dollars, it is 25% that it is tails universe and you found the second dollar of the two.
Note that both 25%+25% give their parent probability 50% of Tails universe, probability of Tails universe matches the probability of Heads universe and these 25% are equal only between themselves, and not to the dollar which is on already divergent branch of probability and not anyhow connected to the others.
In this pancake experiment you throw away half universes where you get a b/g pancake. In my problem you do not throw away anything.
@KongoLandwalker I'm thinking the dollar bill problem isn't quite rigorously defined enough for us to be sure if we disagree. But the reasoning you're using sounds like it could apply to the pancake problem as well. The golden-golden and golden-brown pancakes started out equally likely and seeing a golden side doesn't eliminate the golden-brown possibility so...
It seems similar with the dollar bills. I'd expect finding a dollar bill to be more likely if there are more of them. If so, finding a dollar should increase my subjective probability that I'm in the world where 2 dollar bills were hidden rather than just 1. Aka, I update my prior based on the Bayesian evidence.
I'd expect finding a dollar bill to be more likely if there are more of them.
1) Many dollars -> more chance to find at least one.
(Actually, in my problem you are only asked the question after you find the dollar, so at the moment you are 100% holding a dollar in your hands and the previous statement becomes meaningless and outdated, as you already updated to more recent 100%).
But that scenario you try to convert into
2) found the first dollar -> more chance of many dollars.
There was an A->B which is true (not withing the problem, but in real life situations). And from that you state that B->A is true.
"If A then B" does not mean "if B then A".
*in pancakes there is a possibility of b/g to show up which you didn't truncate in your example. Having appropriate sides does not mean all outcomes are appropriate. According to your problem we look at top side, so the sides are outcomes, not pancakes as a whole.
But in dollar problem in the Heads case there is NO option of NOT finding the first dollar. Nothing to truncate.
"Search" is not in the scope of the problem.
If "finding" is only associated with "search" (your "more money-> more chance" looks like that) then imagine, that the dollars will be in predetermined places. The first in the kitchen on the table. The second if it exists will be in basement. The structure of the house is that you need to pass through kitchen to go to basement.
In the original question you were asked AFTER you find the first dollar, which is equivalent to asking you between kitchen and basement. Note that I only changed where I place the dollars, i did not change the algorithm or the probability tree.
Yay! The market has immediately jumped to the right answer. 🎉
Everyone wants to say 50% at first because it's either the golden-golden pancake or the golden-brown pancake, who knows which?
A clever way to get the right answer: don't think about 3 pancakes in the hat, think about 6 pancake sides in the hat. There are 3 golden sides and you grabbed one of them randomly. But now 2 of those golden sides you might've grabbed are attached to each other. If you grabbed either of them then the plopped pancake's other side will be golden. Only if you grabbed the 1 golden side attached to a brown side will the plopped pancake's other side be brown.
So that reasoning gives the right answer of 2/3.
But that's kind of too clever. How would you know to think about it that way? How can we make it feel true that 1/2 is wrong? Bayesian reasoning! You see this pancake on the table and you think "which world am I in? the golden-golden world or the golden-brown world?". Then you consider the evidence: "I see a golden side". In the golden-golden world, that's fully unsurprising -- you always see a golden side in the golden-golden world. Duh. In the golden-brown world, there's ... not surprise per se but, y'know, not a total non-surprise. Seeing a golden side is only 50% likely in the golden-brown world. So seeing a golden side shifts your probability towards the golden-golden world where seeing a golden side is more likely.
Another intuition from a friend:
You don't have an even chance of getting the g/g and g/b pancakes -- half the time you pull the g/b pancake out, you see the b side, and discard those pulls, just like you're discarding the b/b pancake every time. So the g/b pancake is pulled half as often as the g/g pancake for this situation.