The probability of which coin you have affects the probability of the next toss coming up heads, so having this knowledge is implicit in determining the solution.
That probability was determined the moment you picked the coin out of the jar.
It makes no difference what you do to it after the pick. Hold it in your hand for a day, flip it 10 times, sit on it, whatever - the end result is that p(heads) for that particular coin has not changed. p(heads) will be either 0.5 for a real coin or 1.0 for the rigged one.
The probability then comes down to what coin you picked at the start of the trial. There's a 0.999 chance you have a real one, and 0.001 chance that you have the rigged one.
What if you picked a random coin from the jar, then you looked at it and saw that both sides are heads. Is the probability that this is the coin with both sides heads still 0.001? No, the probability that this is the coin with double heads is 100%.
Now if you pick a random coin from the jar, and you randomly observe one of the sides of the coin 1 billion times and every time you see heads, is the probability that this is the coin with both sides heads still 0.001? No, the probability that this is the coin with double heads is very close to 100%.
Now if you pick a random coin from the jar, and you flip the coin 1 billion times, and every time it comes up heads, is the probability that this is the coin with both sides heads still 0.001? No, the probability that this is the coin with double heads is very close to 100%.
How about if you flipped it 10 times and it came up heads 10 times? Turns out the probability that it is the coin with double heads is about 51%.
Probability quantifies the degree of uncertainty YOU have about the world. This can change even when the world doesn't change, namely when you observe something about the world.
I believe someone already presented this analogy to you, but I'm curious what your response is. Imagine the jar has only two coins, one always heads and one always tails. Choose a coin randomly, then flip it ten times. If you get ten heads, what is the probability that the next flip is heads?
According to the methodology you are advocating, the probability would be 50%, because you are only considering the initial probability of selecting a coin from the jar. But using the methodology I suggested in another comment, you would list out every possible outcome and conclude that there is a 100% chance of getting another heads.
OK, What if I flipped it 10,000,000 times and got all heads? What is the probability that I have the non-fair coin? What is the probability for the next toss?
Another way to look at it - one coin has two heads, the other 999 have two tails. You flip the coin and get one heads. What is the possibility that the next will be heads?
That's not really the problem. I believe I was able to explain the correct response using classical probability: https://news.ycombinator.com/item?id=7001288. This is no more complex than asking what the odds are of being dealt a full house in poker. You can enumerate all equally likely outcomes and simply count them.
