python-dice-top-k-game-probabilities/README.org

#+title: Python Programming Dice Probabilities
#+auto_tangle: t
#+PROPERTY: session *python*
#+PROPERTY: cache yes
#+PROPERTY: exports both
#+PROPERTY: header-args :tangle pydiceprob.py
* Introduction
:LOGBOOK:
CLOCK: [2024-04-19 Fri 10:33]--[2024-04-19 Fri 12:50] =>  2:17
CLOCK: [2024-04-17 Wed 13:40]--[2024-04-17 Wed 14:14] =>  0:34
:END:

I was given a problem as such:

#+begin_quote
You have two players, Bob and Alice, that take turns in rolling a fair k-sided die. Whoever rolls a
k first wins the game. The Python program should output the probability that Bob wins the
game for k = 6 thru 99. That is, the output will be an array of probabilities where index 0 is the
probability when k = 6; index 1 when k = 7; etc.
#+end_quote

Bonus points:
#+begin_quote
If you have 8me and interest, create a REST server rather than a console program. Flask or
FastAPI can be used. The REST endpoint should be a GET, accept no Request Body, accept an
op8onal Header for “k”, and return:
- The array of probabili8es in JSON format if no “k” is provided in the Header
- A single probability in JSON format if a “k” is provided in the Header
#+end_quote

* Thinking
Given the problem's nature, we need the following functionality:

1. Output the probability for Bob to win.
2. Create an array of probabilities that contains the data for the output.
3. Calculate the probabilities based on the value of =k=.
4. Define the value of =k=, between 6 and 99 inclusive.

For the bonus secion:
1. Create a FastAPI endpoint that takes an optional Header for =k=.
2. Return either the full array of probabilities (no =k= provided), or return the
   specific probability if =k= is provided.


* Code
The following is python code blocks, with the documentation attached. Using
=org-babel=, these are tangled into =pydiceprob.py=,
which is the final python script.
** Imports
#+begin_src python 
import sys
import json
import pprint
#+end_src

#+RESULTS:
: None
** Pretty Print
#+begin_src python 
pp = pp.Printer = pprint.PrettyPrinter(indent=2, compact=True)
#+end_src
And a usage printe

#+begin_src python
def usagequit():
    print("""Usage: python pydiceprob.py [k] [mode]
    k between 6 and 99
    modes: single, single-table, multi, multi-table
    single prints out the probability of a single throw with a k-sided dice yielding k.
    single-table prints out a table between 6 and k with the probabilities of a single throw yielding k.
    multi prints out the probability of winning (assuming we're going first) in a game where two players take turns and the person who throws k first wins.
    multi-table prints out the probability of winning for a range between 6 and k if you have the first throw.""")
    quit()
#+end_src
** Main
Main function for managing CLI interactions, and dispatch
#+begin_src python 
def main():
    global k
    try:
        if int(sys.argv[1]) >= 6:
            k = int(sys.argv[1])
        else:
            usagequit()
    except IndexError:
        usagequit()
    global mode
    try:
        mode = sys.argv[2]
    except IndexError:
        usagequit()
    dispatch(mode, k)

#+end_src

#+begin_src python 
def dispatch(mode, k):
    modes = ["single", "multi", "single-table", "multi-table"]
    if mode not in modes:
        usagequit()
    if mode == "single":
        print(first_turn_probability(k))
    elif mode == "multi" :
        print(multi_turn_single(k))
    elif mode == "single-table":
        print(first_turn_protatbilities(k))
    elif mode == "multi-table":
        print(multi_turn_table(k))


#+end_src
** Array of dice and players
Second, we need to grab the value of =k=, the size of the dice we're calculating
the probability for. Depending on the implementation we choose to use, we'll be either
grabbing it from the CLI, or the REST API. For now, this will remain in the CLI.

Then for the bonus code, we need several steps. First, we need to generate a list
such that we have numbers between 6 and 99 in it.
#+begin_src python 
def dice(k):
    if k < 6:
        print("Dice must be at least 6-sided")
    elif k == 6:
        dice_array = [6]
    else:
        dice_array = list(range(6, k))
#+end_src

#+RESULTS:
: None


** Single turn win probability
Then, we have to calculate the probability of a win on each throw given a dice
of size =k=.

Both players (Alice and Bob) throw dice, alternating, and the person who throws the top number first (=k=) wins. Bob always goes first for simplicity.

A dice of size =k= has a 1/k probability of winning each throw.

=k+1= here, because python calculates range(a, b) indices in the way of b-a, so
for a = 6 and b = 10, it'll give an array of 4 items (10-6 = 4) as such:
~[6, 7, 8, 9]~, as it counts the first item as 1.

Using =k+1= we can get the correct, inclusive array that we desire: ~[6, 7, 8, 9, 10]~.
#+begin_src python 
def first_turn_probabilities(k):
    result = {}
    for k in range(6, int(k)+1):
        result[k] = 1/k
    return result

def first_turn_probability(k):
    return first_turn_probabilities(k)[k]

#+end_src

#+RESULTS:
** Multi-turn win probability
Over many throws, the advantage of winning will decrease, approaching 50%.

We're assuming Bob goes first, and then is followed by Alice.
For each throw, the probability of winning is the same.

#+begin_src python
def multi_turn_single(k):
    p_win = 1 / k  # Winning probability on any given throw
    p_lose = (k-1) / k   # Losing probability on any given throw
    bob_wins_prob_sum = 0
    r = p_lose**2
    probability_win = p_win / (1 - r)
    return probability_win
#+end_src

And then we want to generate a table of all the probabilities up to =k=.

#+begin_src python
def multi_turn_table(k):
    result = {}
    for i in range(6, int(k+1)):
        result[i] = multi_turn_single(i)
    return result
#+end_src
** Print
#+begin_src python 
def print(stuff):
    pp.pprint(stuff)
#+end_src

** Script
To run as a script.

#+begin_src python 
if __name__ == "__main__":
    main()
#+end_src