Flip to freedom

This week’s Riddler Classic is a problem about coin flipping. The text of the original problem is quite long, so I will paraphrase it here:

There are $n$ prisoners, each with access to a random number generator (generates uniform random numbers in $[0,1]$) and a fair coin. Each prisoner is given the opportunity to flip their coin once if they so choose. If all of the flipped coins come up Heads, all prisoners are released. But if any of the flipped coins come up Tails, or if no coins are flipped at all, the prisoners are not released. If the prisoners cannot communicate in any way and do not know who is flipping their coin or not, how can they maximize their chances of being released?

Here is my solution:
[Show Solution]

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