Question 96
On floor $0$ of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor $1$. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing.
Between the third and the $100^{\text{th}}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
Proposed by: Kevin Yang
Between the third and the $100^{\text{th}}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
Proposed by: Kevin Yang