Average duration based on changing probabilty

Given a game where the player has some probability of dying and this probability increases each second as the game is played to some maximum probability. The player has three lives and then the game is over.

Can I calculate the average game duration if I know the probability the player will die at each second?

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Yes. If $p_n$ is the probability for the char to die if it is at age $n$, then the porbability of dying at age $n$ is $$(1-p_0)(1-p_1)(1-p_2)\cdots(1-p_{n-1})p_n$$ hence the expected life time is $$\sum_{n=0}^\infty n\cdot (1-p_0)(1-p_1)(1-p_2)\cdots(1-p_{n-1})p_n\\ =\sum_{n=1}^\infty (1-p_0)(1-p_1)\cdots(1-p_{n-1})$$ and with three lives, the expected game duration is three times as long.
If all $p_n$ were equal, this expression could readily be simplified, but alas that does not match your setup.