Rolling Higher

Probability Level 2

Roll a fair six-sided die until you roll a number that is less than one of your previous rolls. To three decimal places, what is the expected value of the number of rolls made?

Bonus: Suppose you rolled a fair $n$ -sided die instead. What is the expected value of the number of rolls made?

Extra Bonus: As $n \to \infty,$ what does the expected value of the number of rolls approach?