Puzzle o' the Day 406!
Sometimes I present a puzzle because I want someone to come up with a more elegant answer than mine. So it is with this puzzle, where my solution is a tangle of algebra that mysteriously resolves into a simple expression. I hope you can do better!
You keep rolling a 6-sided die until you roll a number that's less than your previous roll.
a. (The puzzle: medium.) What's the expected number of rolls?
b.i. (Easy if you got a.)
What's the expected number of rolls for an n-sided die?
b.ii. (Medium-easy if you got a.) What happens to the expected number of rolls as n goes to infinity?
You keep rolling a 6-sided die until you roll a number that's less than your previous roll.
a. (The puzzle: medium.) What's the expected number of rolls?
b.i. (Easy if you got a.)
What's the expected number of rolls for an n-sided die?
b.ii. (Medium-easy if you got a.) What happens to the expected number of rolls as n goes to infinity?