Otherwise roll again, this time you succeed if you roll a 1 or a 2.

Otherwise roll again, this time you succeed if you roll a 1, 2 or 3.

Otherwise continue the pattern until you eventually win.

Let x = number of tries to success. What is the expected value of x?

Extend to an n-sided die and give a formula for E(n,x).

For any value of n there is a maximum k such that P(x≤k)≤1/2. Find a formula for this k in terms of n.

This might make an interesting casino game but I've never seen it before.