Incidentally, I am reminded of [Littlewood’s??] scenario where you take a set of balls labelled 1, 2, 3, … and perform an infinite number of operations: first put ball 1 into an urn and then remove it. In step 2, put balls 2–3 into the urn and remove ball #2. Step 3 is to put balls 4–6 into the urn and remove ball 3, and so on. Step k adds k balls and takes one away, so after any finite number of steps the number of balls keeps growing and growing. Yet ball k is withdrawn in step k so every ball ends up removed!
In any one dice roll, it is true that the player has a 97% chance of winning.
It is also true that, if the player was in some group that played the game, and the game is now over, the player had a 90% chance of being in the losing group, because that group makes up such a large proportion of the overall number of players.
There is no paradox.
Different questions have different answers that depend on different things.
This is the wrong tense to put this in. Only after a game has been played until the end does a randomly selected player of the game have a 90% chance of being in the double six group. As others have pointed out, you cannot gather a group of people and tell them that 1) everyone will be part of a group that is subjected to a roll, and 3) there will definitely be a double six at the end. You cannot guarantee that. So a player does not “have” a 90% chance to be in the double six group. Only after the fact, after playing a random number of games with a random number of total players, can you assign a probability to a member of a group that was subjected to the roll, but you have no idea how large that group would actually become.
In reality, if you tried to calculate an individual’s odds before the game started, the game could either stop before it got to them and conversely you could run out of people before rolling double sixes.
I don’t feel this is true. Saying that a player has a 97% chance of winning is the same as saying a player has a 97% chance of not being in the group that has the double-six dice roll. And that’s not true. Ninety percent of the players will be in the double-six group. So a player has a ninety percent chance of losing and a ten percent chance of winning.
As I interpret it, “in any one dice roll” is essentially the same as “in any one group”; so you can’t then talk about chances of being or not being in a particular group.
I feel that’s what created the apparent paradox. The paradox is resolved by realizing that the players are not individually affected by the dice rolls. The effects of a single dice roll are applied to multiple players, so they are effectively linked.
Think of it as a game being played by dice instead of a game being played by people. Imagine there are eleven teams of dice numbered two through twelve. Each team collects the number of people that were in a group if its number is rolled. Which number ends up with the most people?
It’s clear that, given the way the game works, twelve will end up with the largest collection of people. So now turn it around and ask the people which number’s collection are they most likely to end up in.
But, if the game hasn’t ended yet, and I’m in the group who are about to roll the dice, my chance of winning is 97%. And, that does not create any kind of paradox, no matter how many people are in my group.
But those are the same question. And again, the paradox is because there are assumed to be an infinite number of people. If there are a finite number of people and money, then there’s the possibility that the game ends without anyone losing, because we ran out of money or people first.
While the odds of rolling a double six are only three percent, the size of the group which rolls a double six is nine times larger than all of the other groups combined. So ninety percent of the people in the game will be in the group that has a double six even though that roll is uncommon.
In most of the simple probability examples we encounter, the percentage of individuals falling into a certain category equals the probability of falling into that category:
If I sell 100 raffle tickets and 3 of them are prize-winners,
- 3% of the tickets are winners
- the probability of winning is 3%.
If I pick a card out of a regaular deck of cards,
- 25% of the cards are spades
- the probability of picking a spade is 25%.
Right. So does it follow that the probability of dying—that is, of being a person who gets a double six—is 90%? Yes, if you are careful to specify that what you mean is that the probability of a player, randomly selected from all the players in the game in such a way that each of those players has an equal chance of being selected, being one of the ones who gets a double six is 90%. That’s analogous to the other examples: picking a card means that each card in the deck has an equal probability of being the one you select.
Now what about the 3% probability of rolling a double six? What does that represent 3% of? It’s not 3% of a certain group of players. The only interpretation I can see that makes sense is that it’s 3% of the possible ways a pair of dice can come up. But that’s something completely different from the percentage of players who lose or win, so there’s not really any paradox.
No. I’m talking about an independent roll of the dice. There’s no paradox. The number of players is immaterial if you are looking at an independent roll.
Yes. So when you look at any question that looks at more than one roll, or more than one group, it’s a different question.
But you can’t define the number of people in the game until the double six has been rolled. Until that happens, the potential number of people in the game is unlimited. If you can a priori define the number of players in the game, then for everyone else (which is an unlimited number of people) the risk is 0%.
So exactly who does the probability apply to? You can’t identify who they are until they’re dead, and at that point you know the results.
The only way this information would be useful is if the rolls had been rolled ahead of time in secret and you were informed that you would be subjected to a roll but did not know what that roll would be. Assuming that the rolls and your placement in them were fair, then your odds of dying would indeed be high.
If the rolls and groups had not been determined ahead of time, then there is no way to know if you would not be subject to a roll at all, or were subject to one and survived, or were subjected to one and died, or that the game ended before a double six ever got rolled.
I think the core issue is that we’re looking at it after the fact. The design of the problem makes the percentages undefined until the game ends. For example, let’s say we are all in the fourth group. What are our odds of surviving the round? 97%.
It is only after the fact that we can look back and say that the people in the last group had a 90% chance of being in the losing group. Until it’s over, we do not know how many groups there will be.
I think this was illustrated quite well with the single coin flip. I guess tails, flip one coin once, it comes up tails. I only had a 50% chance of winning, but now after the fact I have won 100% of the time.
I disagree. We may not know the amount of people in the final group but that’s not the number we’re looking for. What we’re looking for is what percentage of people will be in the final group. And the way the game is set let’s us determine that percentage (within a small range) regardless of how many groups there are. We can prove that approximately ninety percent of the total will be in the final group, regardless of whether that final group is the second group, the tenth group, the fiftieth group, or the ten thousandth group. And we can do this with the information we have before the game starts.
That’s not how probabilities work.
What percentage of babies born in the year 2033 will be male? I can predict the probability is fifty-one percent even though none of those babies have been conceived.
But it’s obvious silliness. Consider a single coin flip where if I guess correctly I win a dollar, but if I guess incorrectly a million other people win a dollar. The odds on the coin flip are obviously still 50/50, correct? It doesn’t become a million to one shot just because there are a million people in the other group. You have to ask a completely different, unrelated question to get the million to one shot.
I would think this example would make clear that the two different probabilities are answers to two very different questions. No paradox.
I don’t see how this relates to your previous post where you said the percentages are undefined until the game ends. And that was the point I was arguing against.
Suppose you have an unlimited group of people, and an unlimited supply of money so you can run this experiment. Before you start, is it possible for any person to know what their odds are of surviving the game?