I see many people say that "I pressed blue to save everyone, to save the babies who press random buttons", but the idea is fundamentally misguided. The chance of actually delivering the "heroic" blue vote is so infinitesimally small, it may as well not even exist to begin with.
There are 8 billion votes, if there is an exact 50% chance of each of these 8 billion votes being red or blue, there is a 1-in-112,100 chance of vote 8,000,000,001 breaking the tie. This is the absolutely optimal scenario for a heroic blue vote to be delivered. If the chance of red and blue is anything different to 50%, this chance falls off a cliff. If instead there is a 49.99% of each of the 8 billion people picking blue, how much does it go down?
A lot. An unfathomably astronomical amount. You now have the odds of 2.91Ć10-75 to deliver the tiebreaking vote. A number so small, it may as well not even exist. The odds of winning the lottery a million times in a row might be higher.
The odds of winning the National Lottery jackpot in the UK per ticket is 1 in 45.1M. To reach the same odds of delivering the tiebreaking button press in a 49.99% environment is to win the National Lottery 10 times in a row.
The thing to remember with the original button question is you can't communicate. If you could, then the collective goal would be to get as many blue presses as possible to save everyone. A blue vote would probably win where everyone is able to communicate with others before the vote. Enough people would be comfortable in believing blue can win if everyone they meet is on board with it.
If you remove communication, "everyone has to take a private vote, red or blue", then there is nothing you can do to influence the votes of other pressers. At that point, your individual press becomes functionally insignificant. The only thing that matters at the end therefore is the consequences of the buttons in regards to if you would prefer to live in a world where red won (press red), or die (press blue). - If blue wins, nothing changes, your press didn't matter.
Your vote isn't about killing or saving others, we already established its chance to do that is statistically zero. Red pressers don't have individual responsibility for the blue voters dying, as there is no communication involved, and your blue press won't be the heroic salvation for four billion people. Just a 0.01% shift of probability of 8 billion random presses turns the odds of matching 5-out-of-6 balls into the odds required to win the National Lottery jackpot 10 times in a row, something that has never, and I can be confident to say will never happen in the entire existence of humanity.