.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/classical/plot_01_buffon_needle.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_classical_plot_01_buffon_needle.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_classical_plot_01_buffon_needle.py:


Buffon's needle: estimating pi by dropping needles
========================================================

A needle as long as the gap between parallel lines crosses a line with
probability :math:`2/\pi`, so counting crossings estimates :math:`\pi`.
The estimate's error shrinks like :math:`1/\sqrt{n}`.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import buffon_needle








.. GENERATED FROM PYTHON SOURCE LINES 17-19

One large experiment
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-24

.. code-block:: Python


    result = buffon_needle(n_drops=1000000, seed=0)
    print(f"crossing fraction {result.crossing_fraction:.5f} (exact 2/pi = {result.exact_crossing_probability:.5f})")
    print(f"pi estimate {result.pi_estimate:.5f}")





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

 .. code-block:: none

    crossing fraction 0.63669 (exact 2/pi = 0.63662)
    pi estimate 3.14125




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Convergence with the number of drops
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-38

.. code-block:: Python


    n_values = np.logspace(2, 6, 9).astype(int)
    errors = [np.mean([abs(buffon_needle(n, seed=s).pi_estimate - np.pi) for s in range(20)]) for n in n_values]

    fig, ax = plt.subplots()
    ax.loglog(n_values, errors, "o-", label="mean |error| over 20 runs")
    ax.loglog(n_values, errors[0] * np.sqrt(n_values[0] / n_values), "--", color="0.5", label=r"$\propto 1/\sqrt{n}$")
    ax.set_xlabel("needles dropped")
    ax.set_ylabel(r"error in $\hat\pi$")
    ax.legend()
    ax.set_title("Buffon's needle (1777)")



.. image-sg:: /api/gallery/probability/classical/images/sphx_glr_plot_01_buffon_needle_001.png
   :alt: Buffon's needle (1777)
   :srcset: /api/gallery/probability/classical/images/sphx_glr_plot_01_buffon_needle_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, "Buffon's needle (1777)")




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

   **Total running time of the script:** (0 minutes 0.567 seconds)


.. _sphx_glr_download_api_gallery_probability_classical_plot_01_buffon_needle.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_buffon_needle.ipynb <plot_01_buffon_needle.ipynb>`

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

      :download:`Download Python source code: plot_01_buffon_needle.py <plot_01_buffon_needle.py>`

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

      :download:`Download zipped: plot_01_buffon_needle.zip <plot_01_buffon_needle.zip>`


.. only:: html

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

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