.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/discrete/plot_01_problem_of_points.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_probability_discrete_plot_01_problem_of_points.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_probability_discrete_plot_01_problem_of_points.py:


Pascal and Fermat's problem of points: dividing the stakes
==========================================================

Two players stake equal amounts on a game of fair rounds; the first to
win a set number of rounds takes the pot. The game is interrupted when
player A still needs :math:`a` wins and player B needs :math:`b`. How
should the pot be split?

Pascal and Fermat's answer (1654): imagine the at most
:math:`a + b - 1` remaining rounds are all played. A takes the pot
exactly when A wins at least :math:`a` of them, so A's fair share is

.. math::

   P(\text{A wins}) = \sum_{k=a}^{a+b-1} \binom{a+b-1}{k} 2^{-(a+b-1)},

a binomial tail probability.

.. GENERATED FROM PYTHON SOURCE LINES 22-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import Binomial


    def share_of_a(a: int, b: int) -> float:
        """Fair share of the pot for the player who still needs ``a`` wins (opponent needs ``b``)."""
        remaining = Binomial(n=a + b - 1, p=0.5)
        return float(1.0 - remaining.cdf(a - 1))









.. GENERATED FROM PYTHON SOURCE LINES 35-41

Pascal's example: A needs 1 more win, B needs 2
------------------------------------------------------------------

In his letter of 29 July 1654 Pascal splits a pot of 64 pistoles. With
two rounds left to imagine, B only takes the pot by winning both, so A
is owed 3/4 of it: 48 pistoles.

.. GENERATED FROM PYTHON SOURCE LINES 41-56

.. code-block:: Python


    a, b = 1, 2
    n_left = a + b - 1
    remaining = Binomial(n=n_left, p=0.5)
    ks = np.arange(n_left + 1)
    print(f"A's share = {share_of_a(a, b):.4f} -> {64 * share_of_a(a, b):.0f} of 64 pistoles (Pascal: 48)")

    fig, ax = plt.subplots()
    colors = ["tab:blue" if k >= a else "tab:red" for k in ks]
    ax.bar(ks, remaining.pmf(ks), color=colors)
    ax.set_xticks(ks)
    ax.set_xlabel(f"rounds won by A among the {n_left} imagined remaining rounds")
    ax.set_ylabel("probability")
    ax.set_title("Problem of points: blue = A takes the pot, red = B does")




.. image-sg:: /api/gallery/probability/discrete/images/sphx_glr_plot_01_problem_of_points_001.png
   :alt: Problem of points: blue = A takes the pot, red = B does
   :srcset: /api/gallery/probability/discrete/images/sphx_glr_plot_01_problem_of_points_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    A's share = 0.7500 -> 48 of 64 pistoles (Pascal: 48)

    Text(0.5, 1.0, 'Problem of points: blue = A takes the pot, red = B does')



.. GENERATED FROM PYTHON SOURCE LINES 57-59

Check by simulating the imagined remaining rounds
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 59-64

.. code-block:: Python


    a, b = 3, 5
    wins = Binomial(n=a + b - 1, p=0.5).sample(size=200000, seed=0)
    print(f"A needs {a}, B needs {b}: exact share {share_of_a(a, b):.4f}, simulated {np.mean(wins >= a):.4f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    A needs 3, B needs 5: exact share 0.7734, simulated 0.7738




.. GENERATED FROM PYTHON SOURCE LINES 65-67

The fair division for every interrupted score
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 67-77

.. code-block:: Python


    max_needed = 8
    table = np.array([[share_of_a(i, j) for j in range(1, max_needed + 1)] for i in range(1, max_needed + 1)])
    fig, ax = plt.subplots()
    im = ax.imshow(table, origin="lower", cmap="coolwarm_r", vmin=0.0, vmax=1.0, extent=(0.5, max_needed + 0.5, 0.5, max_needed + 0.5))
    ax.set_xlabel("rounds B still needs, $b$")
    ax.set_ylabel("rounds A still needs, $a$")
    ax.set_title("A's fair share of the stakes")
    fig.colorbar(im, ax=ax, label="fraction of the pot owed to A")
    print("equal needs give an even split:", all(abs(table[i, i] - 0.5) < 1e-12 for i in range(max_needed)))



.. image-sg:: /api/gallery/probability/discrete/images/sphx_glr_plot_01_problem_of_points_002.png
   :alt: A's fair share of the stakes
   :srcset: /api/gallery/probability/discrete/images/sphx_glr_plot_01_problem_of_points_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    equal needs give an even split: True





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.072 seconds)


.. _sphx_glr_download_api_gallery_probability_discrete_plot_01_problem_of_points.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_problem_of_points.ipynb <plot_01_problem_of_points.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_problem_of_points.py <plot_01_problem_of_points.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_problem_of_points.zip <plot_01_problem_of_points.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_