Monte Carlo AI in 21 game

I am very interested in the Monte Carlo AI.

I tried my best, still, this AI plays very badly.

This code "works" in the meaning that it does not crash, but the quality of play is extremely low.

Have I completly misunderstood the Monte Carlo AI or is there just a nasty bug in my code preventing it from playing correctly?

Can you understand every detail of code?

Creating an AI is quite complex a task so I would like to have the better style possible in order to keep the code understandable by everyone.

```
"""
TITLE: Monte Carlo AI that plays the 21 game
AUTHOR: Caridorc Tergilti
LICENSE: Creative Commons 3.0
CURRENT_STATE: The AI is currently very weak, I have no clue why.

General explanation:
This game is very,very easy, an AI can be disigned in
much simpler ways than this.

I designed a Monte Carlo AI because I want to learn about Monte Carlo AI-s
(You can find out more about Monte Carlo here:
https://en.wikipedia.org/wiki/Monte_Carlo_method)
and I want to discover if such an AI can actually be powerful in a game.

(Yes, there are many articles and papers on the internet that say that
Monte Carlo AI works but I want to prove it myself).

This programme is also Creative Commons so I hope that many people will enjoy
reading this programme to understand Monte Carlo AI better.

I will now explain my understanding of Monte Carlo AI.

1) The computer, knowing the rules, must decide what move is better.
2) FOREACH move the computer is going to simulate the move.
3) It will than play a big number of random games starting from the position after he made
the move.
4) Each move will be awarded a score equals to: wins / total_games.
5) The move with the highest score will be played.
"""

import random

WELCOME = """
21 game.
The total starts at 0.
Each player can chose 1, 2, or 3,
the number is than added to the total.
If when a player adds his number,
he makes the total equals or more than 21, he loses.
"""
DEPTH = 1000

def play_random_game(state):
"""
This fun

Your simulation assumes the human is playing randomly. This is not realistic. Consider the case when it's the human's turn at 19. A sensible player will certainly play 1 to force a win, but a random player has only 1/3 probability of doing the same. The simulation teaches the AI that the computer has an advantage in that situation, which is wrong.

If the rules of the game were changed so that every move must be made at random, your algorithm should be able to predict each player's probability of winning at any point. However, that would be a different game: a game of chance.

There's also a simple bug: you generate random moves by randint(0,3). This should of course be randint(1,3) because 0 is not a valid move.

StackExchange Code Review Q#67732, answer score: 4