u/EntireEntity

Humanity is divided into two groups. One group gets the original button problem.

The other group has an additional third option: to just leave. People who leave, will not contribute to the red/blue count.

If blue gets more "votes", everybody survives.

If red gets more "votes", blue dies.

All participants are made aware that the two groups exist and the rules surrounding that.

What do you pick, if you find yourself in the group that receives the standard button problem? Do you pick the same color as in the original version? If you would change the color you pick, why?

What do you pick, if you find yourself in the group that has the extra option to leave? Is there any reason left to pick red? If there is no reason to press red, should you now press blue instead of leaving? Since nobody will reasonably press red in that second group, everyone could just realize that if they press blue, they get 50% blue by default and only one blue vote is required in the other group to have a majority.

So now in the first group everyone could realize that there is no more risk in taking blue, as you can now be certain that 1 blue vote (yours) will be enough to get the majority. Is this line of thought logically consistent? Is a blue majority the logical conclusion of this variation instead of red?

I have read this sentiment a couple of times now "If you vote blue, there is a 1 in 8 billion (≈ population of the earth) chance that you change the outcome, so just press red and survive."

I don't want to discuss whether that is a good reason to press red or not, I am more interested in the statistical correctness of that number. So if you don't care about that kind of discussion, feel free to leave "Who cares about the number, you should choose red/blue anyways." and leave a downvote on this post.

I also am not going to provide an answer to the question, as I don't have the statistical skills or knowledge to do so. I will hope that the Reddit collective will be much much smarter than me and answer the question for me.

So, why do I think that it's actually not a 1 in 8 billion chance that the result would be exactly 50/50 without "your" vote?

To me that implies that each of the 8 billion possible outcomes is equally likely, their distribution is uniform and that it could be 100/0 with the exact same probability as 50/50. Just intuitively, I would say that a 100/0 result is practically impossible, whereas 50/50 not as unlikely.

My first thought to find a mathematical explanation was to think about, what would happen, if I had to predict, whether any individual person will press red or blue. Imagine, someone collected 100 people and sat them in front of those two buttons, now you have to blind guess, what they will press. I assume, and this is pulled completely out of thin air, that you would maybe manage to predict around 50% correctly. So, here is my first non-sequitur: There is an inherent 50% chance that a person is more inclined to press red or blue.

So, now that we "know" that any person will press red or blue with a 50% probability, we can think about, the probability of different outcomes. The easiest to calculate are the very edge cases, as there are the fewest permutations. A 100/0 result for example than would have a chance of 0.5^8B, which is some small number X. Now the chance of exactly one person pressing a different button is the same 0.5^8B, BUT multiplied with 8B, since there are 8B possible permutations to get this result. So this result is 8B times more likely than the first. I'll leave the calculation of the remaining 8B - 2 possible outcomes as an excercise to the reader.

Instead, I will just say that with this 50% assumption, we would see a probability distribution of the outcomes, closely resembeling a gaussian. And that out of all the 8 billion possible results, the 50/50 outcome would have the highest individual probability. So... since the most likely individual outcome actually is 50/50, your vote most likely matters? Not exactly, it is still much more likely that the result will not be 50/50 than that it will be exactly 50/50, but it may not be a 1 in 8 billion anymore, so that's at least something.

Now, here are my own issues with my process here, first, the assumption I made. It's likely wrong. In reality we may not see an exact 50% chance for either color. Still, you could just take the exact real numbers and do the whole probability distribution thing and it would turn out that the most likely result will reflect the numbers you plugged in (i.e. if you assume 70% press red, the outcome distribution will also have its maximum at 70% red). You would still see that not all outcomes are equally likely.

Another thing that annoys me, is that it seems wrong to assign probabilities to individuals' button pressing behavior. In reality a person makes an informed decision and doesn't roll an internal die to see, which button to press... or do they? Is the internal die all the experiences that led up to the moment in which they make the choice? What does that say about free will?

And is the probability of correctly predicting that a person will press a particular button actually useful here? I have no idea if statistics change, when you do look at the entire population, instead of just a sample of a population. Can we even assign a probability to the outcome at all?

There are probably some other issues as well that I can't even begin to think about, because of my lack of knowledge in the field. Anyways, if you think, you know the answer to the initial question and want to take the time to explain or at least hint at, how to answer the question, I'd really appreciate it.

Last session I upcasted Invisibility.

>A creature you touch has the Invisible condition until the spell ends. The spell ends early immediately after the target makes an attack roll, deals damage, or casts a spell.

Using a Higher-Level Spell Slot. You can target one additional creature for each spell slot level above 2.

One of the targets wanted to become visible and made an attack roll. Does the other target become visible as well as "The spell ends early immediately after the target makes an attack roll [...]"?

Or does it only end for the target that made the attack roll? And the main description is just worded for one target, but should be interpreted for each target individually?

Is there any official paragraph on this?