Excellent excellent!

If 6 is rolled, then P(X|R=6) = (N-1 choose 9)/(N choose 10)

Might as well reduce that to 10/N to make the rest of the lines easier to read.

If you don’t flip it, you have a 2/3 chance of dying.

There is also a chance that your switch is not connected and someone else has control of the real one. So there is an implicit assumption that everyone else is equally logical as you and equally selfish/altruistic as you, such that whatever logic you use to arrive at a decision, they must have arrived at the same decision.

No matter what your goal is, given the information you have, flipping the switch is always the better choice.

That is my conclusion too! I was surprised to learn though in the comment thread with @pancake that the decision may be different depending on the percentage of altruism in the population. E.g. if you are the only selfish one in an altruistic society, you'd benefit from deliberately not flipping the switch. Being a selfish one in a selfish society reduces to the prisoner's dilemma.