We are given a *multiset* $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$

- two
*distinct instances*$x_r$ and $y_r$ of $M$'s numbers are selected uniformly at random from $M$ (which implies that they cannot be the*same instance*of any number contained in $M$, viz., $x_r$ is selected and temporarily removed from $M$, thereafter $y_r$ is selected from $M\setminus\{x_r\}$ without removing it, and finally $x_r$ is added back to $M$), and - $z_r=\frac{x_r+y_r}{2}$ is added to $M$.

**Question:** What is the probability $p_{r,{\epsilon}}$ that we have $\left|z_r-\frac{1}{2}\right|\le\epsilon$ for a given $\epsilon\in\left(0,\frac{1}{2}\right)$?

**Edit:***I show below the experimental results for different simulations of the random process run with an increasing total number of rounds, and for a single simulation of the random process keeping track of the evolution over time of the total average and the last element added - For the sake of convenience, by writing "rounds", here I am counting the initial insertion of both $0$ and $1$ (simultaneously) as the very first round:*

New simul. with 2^1 rounds - Avg: 0.5 | Last added: 0.5

New simul. with 2^2 rounds - Avg: 0.5 | Last added: 0.5

New simul. with 2^3 rounds - Avg: 0.472222 | Last added: 0.5

New simul. with 2^4 rounds - Avg: 0.430147 | Last added: 0.375

New simul. with 2^5 rounds - Avg: 0.413826 | Last added: 0.40625

New simul. with 2^6 rounds - Avg: 0.40012 | Last added: 0.40625

New simul. with 2^7 rounds - Avg: 0.38313 | Last added: 0.46875

New simul. with 2^8 rounds - Avg: 0.377516 | Last added: 0.378906

New simul. with 2^9 rounds - Avg: 0.366866 | Last added: 0.378906

New simul. with 2^10 rounds - Avg: 0.362607 | Last added: 0.342743

New simul. with 2^11 rounds - Avg: 0.360595 | Last added: 0.358353

New simul. with 2^12 rounds - Avg: 0.359471 | Last added: 0.343569

New simul. with 2^13 rounds - Avg: 0.364962 | Last added: 0.336161

New simul. with 2^14 rounds - Avg: 0.500135 | Last added: 0.497771

New simul. with 2^15 rounds - Avg: 0.49995 | Last added: 0.488623

New simul. with 2^16 rounds - Avg: 0.602851 | Last added: 0.590848

New simul. with 2^17 rounds - Avg: 0.376087 | Last added: 0.372888

New simul. with 2^18 rounds - Avg: 0.655107 | Last added: 0.62898

New simul. with 2^19 rounds - Avg: 0.182425 | Last added: 0.201142

New simul. with 2^20 rounds - Avg: 0.709139 | Last added: 0.713385

New simul. with 2^21 rounds - Avg: 0.219937 | Last added: 0.220374

New simul. with 2^22 rounds - Avg: 0.112707 | Last added: 0.112427

Same simul. r=2^1 - Avg: 0.5 | Last added: 0.5

Same simul. r=2^2 - Avg: 0.5 | Last added: 0.75

Same simul. r=2^3 - Avg: 0.545139 | Last added: 0.46875

Same simul. r=2^4 - Avg: 0.625 | Last added: 0.28125

Same simul. r=2^5 - Avg: 0.60393 | Last added: 0.59375

Same simul. r=2^6 - Avg: 0.568329 | Last added: 0.71875

Same simul. r=2^7 - Avg: 0.57769 | Last added: 0.697266

Same simul. r=2^8 - Avg: 0.573474 | Last added: 0.631714

Same simul. r=2^9 - Avg: 0.575036 | Last added: 0.576538

Same simul. r=2^10 - Avg: 0.577153 | Last added: 0.47583

Same simul. r=2^11 - Avg: 0.578355 | Last added: 0.617221

Same simul. r=2^12 - Avg: 0.576684 | Last added: 0.57461

Same simul. r=2^13 - Avg: 0.576757 | Last added: 0.581285

Same simul. r=2^14 - Avg: 0.577305 | Last added: 0.546254

Same simul. r=2^15 - Avg: 0.577683 | Last added: 0.592735

Same simul. r=2^16 - Avg: 0.577662 | Last added: 0.56319

Same simul. r=2^17 - Avg: 0.577692 | Last added: 0.576607

Same simul. r=2^18 - Avg: 0.577675 | Last added: 0.571428

Same simul. r=2^19 - Avg: 0.577657 | Last added: 0.572818

Same simul. r=2^20 - Avg: 0.577655 | Last added: 0.579482

Same simul. r=2^21 - Avg: 0.577652 | Last added: 0.575974

Same simul. r=2^22 - Avg: 0.577654 | Last added: 0.5777

Same simul. r=2^23 - Avg: 0.577659 | Last added: 0.585123

Same simul. r=2^24 - Avg: 0.577657 | Last added: 0.571693

Same simul. r=2^25 - Avg: 0.577659 | Last added: 0.579782

Same simul. r=2^26 - Avg: 0.577659 | Last added: 0.574194

Same simul. r=2^27 - Avg: 0.577659 | Last added: 0.579098

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