MathDB

Initially, the integer

80

is written on a blackboard. At each step, the integer

x

on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless

x=0

, in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let

P(a,b)

be the probability that after

a

steps, the integer on the board is

b

. Determine

\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}

(that is, the value that the function

\frac{P(a,80)}{P(a,2024)}

approaches as

a

goes to infinity).

Problem Statement