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This is only true if the gambler does not vary the size of his bet.

And this is why the house defines rules on:

* Minimum bet size

* Maximum bet size

* Whether they'll let you change bet sizes during play

Correct if the gambler's money can be divided infinitely small. But if there is a lower limit, like 1c, then ruin is inevitable.

heat death is also inevitable.

the mean time before ruin is also important