Apparently this title "appears subjective", so I'll try to make it as objective as possible.
Suppose you have a fair 2-sided coin which, when flipped, yields Heads or Tails with 50% likelihood each. When you flip a coin and you get Tails, the coin disappears. When you get Heads, the coin turns into two identical coins that work as described above. You start with one coin, and if you ever run out of coins the game ends. As long as you have coins, you will keep perpetually flipping coins until you run out.
Obviously, there is always a chance the game will end, since you can always have a run of Tails and lose all your coins. However, as you get more and more coins, this probability diminishes. What is the probability that the game will end, starting with one coin?
I'm fairly certain the answer is 1, purely by the fact that the expected number of coins you gain each turn is 0. Intuitively, you are bound to have a stroke of bad luck at some point in the infinite span of the game, losing all your coins. I would be interested in seeing if this reasoning is valid.
Moreover, I'm interested to know: What would happen if Heads causes the coin to become three coins? In this scenario, you have an expected gain of 1 coin per flip. What's the probability you'll run out of coins? It's definitely greater than zero, but is it less than one?
Thanks for your help.