**Let P(k) be a random draw of integers between 1 and k (inclusive). Assume you repeatedly apply P, starting at 10^1000. What's the expected number of repeated applications until you get 1? Why?**

What is your understanding of the following statement?

*Assume you repeatedly apply P, starting at 10^1000.*

1) We are asked to repeatedly apply P(k), where k is given and equal to 10^1000.

- 1a) Each drawn element is redrawable again even after being drawn (i.e. each draw is independent)
- 1b) Each drawn element is undrawable after being drawn once (i.e. each draw is dependent on previous draws)

Case 1a)

Since best case is that it takes just one draw and worst case is that it takes k draws, I'd guess the expected number of repeated applications until you get 1 is k/2 but I can't really explain why.

Case 1b)

My understanding is that the probability of getting 1 is 1/k on 1st draw, 1/(k-1) on 2nd draw, ..., all the way up to 1 on kth draw. But how does this help me get to the expected number of repeated applications until you get 1?

Case 2)

Am I overcomplicating it?

Thank you for your hints