.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/inclusion_exclusion/plot_01_hat_check_problem.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_combinatorics_inclusion_exclusion_plot_01_hat_check_problem.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_combinatorics_inclusion_exclusion_plot_01_hat_check_problem.py:


The hat-check problem: derangements via inclusion-exclusion
==================================================================

``n`` people check their hats and receive one back at random; what's
the probability nobody gets their own hat back? The answer converges
to :math:`1/e` remarkably quickly.

.. GENERATED FROM PYTHON SOURCE LINES 11-15

.. code-block:: Python

    import math

    from mathematicskit.combinatorics import derangement_count








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Probability of a total derangement, for increasing n
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python


    for n in range(1, 11):
        d_n = derangement_count(n)
        probability = d_n / math.factorial(n)
        print(f"n={n:2d}: D_n={d_n:>10}, P(derangement) = {probability:.6f} (1/e = {1 / math.e:.6f})")




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

 .. code-block:: none

    n= 1: D_n=         0, P(derangement) = 0.000000 (1/e = 0.367879)
    n= 2: D_n=         1, P(derangement) = 0.500000 (1/e = 0.367879)
    n= 3: D_n=         2, P(derangement) = 0.333333 (1/e = 0.367879)
    n= 4: D_n=         9, P(derangement) = 0.375000 (1/e = 0.367879)
    n= 5: D_n=        44, P(derangement) = 0.366667 (1/e = 0.367879)
    n= 6: D_n=       265, P(derangement) = 0.368056 (1/e = 0.367879)
    n= 7: D_n=      1854, P(derangement) = 0.367857 (1/e = 0.367879)
    n= 8: D_n=     14833, P(derangement) = 0.367882 (1/e = 0.367879)
    n= 9: D_n=    133496, P(derangement) = 0.367879 (1/e = 0.367879)
    n=10: D_n=   1334961, P(derangement) = 0.367879 (1/e = 0.367879)





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

   **Total running time of the script:** (0 minutes 0.000 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_inclusion_exclusion_plot_01_hat_check_problem.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_hat_check_problem.ipynb <plot_01_hat_check_problem.ipynb>`

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

      :download:`Download Python source code: plot_01_hat_check_problem.py <plot_01_hat_check_problem.py>`

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

      :download:`Download zipped: plot_01_hat_check_problem.zip <plot_01_hat_check_problem.zip>`


.. only:: html

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

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