About 1% no matter the probabilities.
Interesting answer from Niclear Hoagie, didn't knew about those micromorts before. Not to support it or expand, but a different angle on the situation, and similar tendencies as end result is a coincidence.
Let's take look at other way to investigate the situation based on stats we know, but before that let's makeup an example to help us for to know what to look for later on.
Imagine we negotiated, successefully, with a million people to help us with a research and conduct few tests. We gather that crowd, in some place, they are together. People aren't offered any strong revards so as there is no punisment, they are free to do what they think they like to do - they are just reluctantly agreed to help us in conducting that test.
We ask the crowd make a step to the left. What are the expected results, many or most of them do step to the left, as they agreed to help us with test, but it also expected that some will stay, or step to the rigth or forward or step to the back. Reasons for those who do not do what we did ask them for may be different, and may be unique for each person, and all those reasoning or curcumstance may be/could be clustered - like some did not obey as a joke(funny or not), some out of principles to do opposite(for many different reasons, including conspiracies), some didn't heard correctly, some misunderstood or made an error(brain missfire), some for the reasons to see what would happen if they do differently(to what they asked to do)
Example is made up, but there are stats to confirm that it likely it all will happen the way I describe, but before some of those stats let's see why it is a fundamental thing.
Evolution
Making things differently than majority is one of the evolution adaptations of ours. And if we dive millions years ago, in time of ancestor species there was a time when not using stick and stones as weapons was the norm, but to change that some had to use them first, not like constant users but more like more often than the others to begin with and continue to be that selected group to do it more often than others for few million or more years until it becomes a tool.
Some had to start breaking/chipping stones, when majority didn't do that.
It is not glorius peoneering or research spirit of humany or adventure which often(often enough in some cases) used as if it is some indespensable quality of humans.
Sticks, stones, fire and using tools in general, before it became a norm was a result of different chaotic(random) attempts to do different things, many many million years of failed attepts to do different stuff not understanding if one better than another, as there wasn't enough brains for all that. Typical case of chaos monkies typing some work on typewriters.
There are two things - keeping the order in any form like traditions and what we know how to do, and opposite of that in form of doing atypical things for to find new ways in the unknown in hope to expand what we know. That chaotic research, that monte carlo method it can not be a majority of efforts and resources, but without it a system can't move further, can't creep on the unknown and make it known thus expanding the known. I mean back then there wasn't enough brains, so there wasn't scientific methods, etc etc etc, it was a different time. But that small fraction of doing against order was the main way to get new stuff. Most of those didn't get useful things, it like those useless inventions in 60-70's, or dot com bubble, or similar stuff but with much worse efficiency.
It took few million years to get to stone tools, and after grasping that tools are good stuff and the way to go, and while developing brains and all that subsequent achievement were faster and faster, but still thanks to certain breaking the order stuff. Over time it became less chaotic, not so long wgo actually, and now it less chaotic, even if it seems the thing is back with generative designs and genetic algorithms, it just our tools do that monkey typewriting instead of us(but still there is enough of that on our part as well).
So some fraction of people which do things against common sense, or against current order and stuff is one of the evolution inheritances we have. Some of those do achieve some good stuff, but majority of them does not, but they all are part of one of survivial strategy and if there would not be those who try, there would not be those who succeed.
today we have more brains, and more ways and tools which help us to expand our capabilities, which may work different or similar to that, but just to see how important it was at some point and how deep at our core it was and still is.
people do that breaking the order not necessarly due the reasoning, it has to be understood, all kinds of brain deviations which we can diagnose, are also part of all that. It like mutations in genetics, not necessarly a good stuff, but on a big number of events over time it is the most basic way to go further, when mutations are a fraction or "normally" working stuff. (When fraction is too big, a system just crumbles)
it all was not necessarly a feature which we got in the process, but more like a flaw which there is no reason to fix to 100%, nor is it easy to fix. So it a flaw, but a flaw we benefit from, from times to times. (It could emerge as feature as well, but, for reasons, I think it wasn't)
Real world examples
That goind against order thing, it has many shapes and forms, so as that urge/reasoning/motivaton/no reasoning/incapacity/disability it has different strength in individuals, affects their action with diferent strength.
It also needs to understand that "breaking the order" it is different from the breaking the law, and is all encompasing - red clothing against all else, fashionable things against all other, all cotton vs fashion, no animal skins, suit vs informal things, apples against mango, city vs suburbs, etc etc. All those things can have additional values like convinience, or increasing decreasing sucesseful mating etc but at core of it is to do something others do not do for to get some sucess, to get some positive revard for that individual.
Breaking the law is one of the things we do have some stats, and it is subpart of that breaking the order stuf, and it are all those who failed sufficiently for us to have them in stats.
https://en.m.wikipedia.org/wiki/Incarceration_in_the_United_States
- In 2018 in the US, there were 698 people incarcerated per 100,000;
- Additionally, 4,751,400 adults in 2013 (1 in 51) were on probation or on parole.[2] In total, 6,899,000 adults were under correctional supervision (probation, parole, jail, or prison) in 2013 – about 2.8% of adults (1 in 35) in the U.S. resident population
It quite significan portion of popupation, of all kinds of risk takers for all kinds of gains. Sure that number is accomulated over the years, not that each of them each day ready to make a risk descision, so as they do not necessarly risk the whole life, but more like a portion of it for the most of them. But it also not all of them, some aren't caugth, some are dead before they caugth. So portion of hardcore risk takers is there, I mean potencial is there.
Most importantly it just one slice of risk takers, I'm not sure on validity of information, but a comment from @Robert Rapplean says there are about 5 million gambling addicts in usa, and this link says it "As many as 10 million Americans live with a gambling addiction.". Again it accomulated over the years number and there sure is variety of hardcoreness, and it correlated to crimes, but it not 100% overlap.
But it again an example of doing something meaningless for hope to get something(a form of breking the order) and sure it has secondary fluff about meaning, but casio always win, so u know. But overall the number is big enough, and potencial is there, which same attribute being less dependant on the probability, less dependant on probabilities than in micromort answer reasoning.
https://en.m.wikipedia.org/wiki/Suicide_in_the_United_States
- In 2018, 14.2 people per 100,000 died by suicide, the highest rate recorded in more than 30 years.
Those guy stake it all, they do not need your million, they just need your button. And dead it is a sucessefult case, it does not include attempts, which is at least few times bigger number for variety of reasons.
Conclusion
There are 3 examples, but it could be more of examples of less extreame cases or more extreame cases(like everest), but my point is that probability does not matter that much as we could expect it from micromort answer. Whatever challenge you throw at people there will be different groups of people with different reasons(or absence of reasons) to accept it.
- whatever challenge there is there will be group of people to respond to it, and size of that group does not depend that much on the challenge itself, its nature or probabilities. For reasons of evolution process it has to be more or less uniform distribution.
Micromort answer is perfectly fine for smaller probabilities, and what small is - chances like 6 in 100'000 (homicide rate) ppl consider it safe enough (it more like 0.0001% per day)
(Mean taking micromort directly for now and here may require some recalculation)
- As illuminated in a 2019 report from the National Safety Council, the lifetime odds of an American dying in a car wreck are roughly 1 in 107. That means that every person in the country with a driver's license and a functional vehicle has about a 0.91% chance of ending up as a victim of a driving-related accident. (source)
All those stats are not an answer to OP's question, and just an attempt to illystrate that for different probabilities there are people with different reasoning who may choose for the win, and even for extreame probabilities, 100% or 0% there will be a group of people to press the button.
Unfortunatly all those stats are accomulated percentages, like that 0.91% chance to die(?) in a car crash in a lifetime, but for each day this probability is low, more like 1 micromort or less.
Another problem with your setting is that you set a condition taking descision now or never, and maybe it is exactly what you need to maximise the button pressing, no matter the probability, like 1-5 second to take a decision before doubts or reasoning kicks in. But because of that it hard to tell what kind of percentages to expect even if we buy that there about 1% of potencially exreame risk takers or order breakers.
But for sure there will be some consistent stat for all that, which will vary like from country to country, or depend on how things are presented or conducted, etc etc but for sure there will be some repeatable percentages, around some number.
I think that giving no time to consider cons and pros, press the button in 5 seconds or never, and presenting it as @Tortliena said "win 1kk" is the way to maximise things. But probability itself not necessarly plays such an important role, if you give that thing a time a year then probably 0.1% is expected, a lifetime opportunity few percent, a day to think about it 0.02 - 0.05%, 5 seconds when just survivial things at play idk 50/50(?), needs to digg what is a percentage between figth and flee reactions, sure not 50/50 but percent(s) maybe.
So all in all it focuses on mechanics, and calculating a number like in the micromort answer may be the way to go to estimate the number, but it just part of whole picture, but often there is a 1% of people to respond to all kinds of challenges, this q by itself an example, 200 views 2 answers (L Dutch does not counts, he works for his rep))))
1% is more abstract number of a summ of all kinds of risk takers we can distinguish, which may be the potencial max if all done perfectly - right words, rigth time - to mobilize those order breakers with 100% efficiency. The situation itself, the button is out of order, and mostly those who have potencial to be order breakers can respond to it, others will cave in. So it more like a thoretical limit, which will be improbable to cross no matter what you do in this situation. (Sure it hypotises level at best,and not a strict number - as "save the planet press the button" there are other things at play here, correlated to order breakers but different)
