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.
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.