Some stray thoughts about Probability.
I had never seen the idea that “you could flip a coin forever and never get heads”.
The truth of that leads to some interesting paradoxes and maybe some precision on what is “possible”.
In an actual (instantiated) beginningless infinity, there really cannot be any unfulfilled possibilities.
However, one paradox that was already mentioned was “could you toss an infinite number of tails?” If that was “possible” then the idea would say “yes”. No matter how slight a chance, this would happen. There are more problems. “How about one million tails, one heads, then one million tails, for an infinite time”. Again, if “possible” then yes. Therefore, since there is an infinite number of possiblities, every infinite possibility would occur. But that’s not possible because you could never fulfill an infinite number of possiblities.
Here’s where this doesn’t work though.
Probability is used in mathematics as a means of prediction.
In this case, we have time and opportunities.
The odds are small that you’d always get Tails. But since it could happen, then it would. We say it “would” because it is considered “possible” – in other words, under some real circumstances, it would necessarily happen.
But the missing factor here (aside from the impossibility of measuring or observing what happens in an infinite beginningless series) is … “randomness”.
Flipping the coin is considered a random event. That’s why probability is used as a measure. If it was 100% certain, then it would be a determined event.
Randomness can be decided before the event or interpreted after the end of the event.
“I flipped the coin fairly and randomly”. That’s the assumption.
“I shuffled the cards fully, so they have a random distribution”. So, beforehand, “random” would mean, paradoxically, an unpredictable “arrangement” which we use probability to predict.
However, to review the results of the coin-flip or the card-shuffle afterwards, we also assess the randomness of the flip.
If the coin came up Tails a million times in a row – this exceeds what the probability would be, given random tosses.
If, even moreso, the coin “eternally” came up Tails – this would mean, necessarily, that it wasn’t a random flip.
In order for the process to be random, different results must be possible and must actually occur.
So, it’s strange, but the odds are 50% on a coin flip. However, if the flip is random, it must be 100% certain that both heads or tails will show up eventually – necessarily in an infinite number of flips.
