I'm going to chime in here -- mostly cuz this is a question I actually have the expertise (such as it is) to answer, which is really cool -- and offer a slightly different perspective from the other responses, one that I've found very useful in the past (especially when faced with complex probabilities)
I'm going to involve multiple universes. (Stephen Strange may or may not be responsible.)
### You draw a single card from a freshly shuffled deck. What is the probability that you drew the Ace of Spades?
Most people can see, pretty clearly, that there are 52 cards, only 1 is the Ace of Spades, your chance is 1 in 52.
However, another way to look at is like this: There are 52 universes, and in each one, you drew a different card. Since only 1 of those universes has you draw the Ace of Spades, there is a 1/52 chance that you are in that universe.
Now consider this:
### You draw a single card from a freshly shuffled deck. You show it to someone else and they say it's an Ace. What is the probability that you drew the Ace of Spades?
There are still 52 universes, where 52 different yous all drew 52 different cards.
However, as soon as someone tells you that you drew an Ace, _you can rule out 48 of those universes_, the ones where where you drew 2, or 3, or King, or whatever. Sure, they may have happened, but _you know you're not in any of those_. There are now only **four** possible universes you could be in, one for each of the suits.
And indeed, if you do the calculations, you will find that in this situation, you have a 1/4 chance of having drawn the Ace of Spades.
We can now apply this technique to your coin flips: You flipped a fair coin nine times and got T all nine times. What is the chance that your next coin flip is T?
The chance is still 1/2 (aka 50-50). Sure, it was really unlikely that you got 9 in a row, but look at the big picture. For 9 flips:
- 510 other yous got a much less interesting sequence of flips and are now posting on r/learnmath about the Monty Hall problem
- 1 other you got heads 9 times in a row and posting to r/learnmath asking if the chance getting heads again is really 50%
Does that help?