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Omni-MATH / On floor 0 of a weird-looking building, you enter an elevator that only has one…

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problem

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 XY\frac{X}{Y}, where XX is your current floor, and YY 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 100th 100^{\text {th }} press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
Plain-text mathematical notation (without MathML)
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 (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^(th ) press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
Original LaTeX notation
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?

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Source and history

task_id
bfdb37f2-a329-5402-9ea2-450b7b5fc595
task_key
test--bfdb37f2-a329-5402-9ea2-450b7b5fc595
task_revision_id
1
config / split
/ test

Official source

initial import