Looks like we are talking past each other, so the only way to continue is to show the calculation:
yours:
P(you die in this round| you play this round) = 1⁄36
your mom’s:
P(you die|game is over) = P(you play in the last ever round) = 9⁄10
You can express P2 by summing P1 over multiple rounds, weighted by the odds of the round being last and by the odds of you playing in it. But the important point that P1 and P2 are probabilities of different events.
Looks like we are talking past each other, so the only way to continue is to show the calculation:
yours:
P(you die in this round| you play this round) = 1⁄36
your mom’s:
P(you die|game is over) = P(you play in the last ever round) = 9⁄10
You can express P2 by summing P1 over multiple rounds, weighted by the odds of the round being last and by the odds of you playing in it. But the important point that P1 and P2 are probabilities of different events.
And with that I am disengaging.