{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Binomial Examples"
      ],
      "id": "c01710ce-950b-4a31-bd67-441c77088838"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import scipy as sp"
      ],
      "id": "a753f75f-181c-4b14-9cda-d5e257e4939f"
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "8.065817517094388e+67"
            ]
          }
        }
      ],
      "source": [
        "sp.special.factorial(52) # https://czep.net/weblog/52cards.html"
      ],
      "id": "d7840d1f-9ca4-4d61-8ec6-4c62d3fd72ab"
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "276.0"
            ]
          }
        }
      ],
      "source": [
        "sp.special.comb(24, 2)"
      ],
      "id": "60550080-21c8-4be4-95d1-61871c2b6d9f"
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {},
      "outputs": [],
      "source": [
        "B = sp.stats.binom(24, 0.1) # Binomial(K = 24, p = 0.1)"
      ],
      "id": "0d71a548-5f85-4db0-8839-405e045c348a"
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "2.4000000000000004"
            ]
          }
        }
      ],
      "source": [
        "B.mean()"
      ],
      "id": "a21d8c41-984a-45ea-9230-08c7f6ce0847"
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "array([3, 2, 1, 2, 3, 3, 2, 5, 5, 3])"
            ]
          }
        }
      ],
      "source": [
        "B.rvs(size = 10) # random variables"
      ],
      "id": "14308e32-6e02-4a39-87e0-9ad1d987e678"
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {},
      "outputs": [],
      "source": [
        "S = np.arange(25) # K + 1"
      ],
      "id": "598c1ece-6387-4c92-a6f1-ca792b021ddc"
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "array([7.97664431e-02, 2.12710515e-01, 2.71796769e-01, 2.21464034e-01,\n",
              "       1.29187353e-01, 5.74166014e-02, 2.02021375e-02, 5.77203929e-03,\n",
              "       1.36284261e-03, 2.69203479e-04, 4.48672465e-05, 6.34486314e-06,\n",
              "       7.63733526e-07, 7.83316437e-08, 6.83847683e-09, 5.06553839e-10,\n",
              "       3.16596149e-11, 1.65540470e-12, 7.15298328e-14, 2.50981870e-15,\n",
              "       6.97171860e-17, 1.47549600e-18, 2.23560000e-20, 2.16000000e-22,\n",
              "       1.00000000e-24])"
            ]
          }
        }
      ],
      "source": [
        "B.pmf(S) # density function (probability mass function)"
      ],
      "id": "7d08471b-50b5-4c2f-9053-475b7183e6b8"
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "1.0"
            ]
          }
        }
      ],
      "source": [
        "np.sum(B.pmf(S))"
      ],
      "id": "e762ca1a-22a4-4d06-9403-f0c27629867a"
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjEAAAGdCAYAAADjWSL8AAAAOnRFWHRTb2Z0d2FyZQBNYXRwbG90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\n"
          }
        }
      ],
      "source": [
        "B = sp.stats.binom(24, 0.5);\n",
        "plt.scatter(S, B.pmf(S));"
      ],
      "id": "27fda01e-0ef5-465a-92ea-e6c5133012d3"
    },
    {
      "cell_type": "code",
      "execution_count": 23,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "array([5.96046448e-08, 1.43051147e-06, 1.64508820e-05, 1.20639801e-04,\n",
              "       6.33358955e-04, 2.53343582e-03, 8.02254677e-03, 2.06294060e-02,\n",
              "       4.38374877e-02, 7.79333115e-02, 1.16899967e-01, 1.48781776e-01,\n",
              "       1.61180258e-01, 1.48781776e-01, 1.16899967e-01, 7.79333115e-02,\n",
              "       4.38374877e-02, 2.06294060e-02, 8.02254677e-03, 2.53343582e-03,\n",
              "       6.33358955e-04, 1.20639801e-04, 1.64508820e-05, 1.43051147e-06,\n",
              "       5.96046448e-08])"
            ]
          }
        }
      ],
      "source": [
        "B.pmf(S)"
      ],
      "id": "c528122d-4e6c-4a47-9adf-cb1807f79354"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "From [Binomial webpage](https://roualdes.us/lecturenotes/binomial)"
      ],
      "id": "0fb19a8a-efb9-4e99-ab61-39e2e37c2694"
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "# 1\n",
        "X = sp.stats.binom(10, 0.95)"
      ],
      "id": "5e4ddc0c-14f3-4bee-b023-848a9aa058cf"
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.5987369392383787"
            ]
          }
        }
      ],
      "source": [
        "X.pmf(10) # density function"
      ],
      "id": "b598d841-a8c7-4078-9177-8b8bf9a5b17e"
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.5987369392383787"
            ]
          }
        }
      ],
      "source": [
        "0.95 ** 10"
      ],
      "id": "7479f573-e0df-483b-a961-f12b5d05271f"
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.07463479852001967"
            ]
          }
        }
      ],
      "source": [
        "X.pmf(8)"
      ],
      "id": "e30e415a-3be7-410a-b887-c77e75108272"
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [],
      "source": [
        "# 2\n",
        "X = sp.stats.binom(50, 0.1)"
      ],
      "id": "f9d1d176-74ed-42da-b65f-ee8232ca5fa6"
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.1849246008952154"
            ]
          }
        }
      ],
      "source": [
        "# 2a\n",
        "X.pmf(5)"
      ],
      "id": "b2f190d5-8414-450b-8892-7d7ec5cf7ab9"
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "array([0.1809045 , 0.1849246 , 0.15410383])"
            ]
          }
        }
      ],
      "source": [
        "X.pmf(np.arange(4, 7))"
      ],
      "id": "37b1e60c-5dcd-43f3-97fd-5a98e50b6534"
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.8882712436536537"
            ]
          }
        }
      ],
      "source": [
        "# 2b\n",
        "np.sum(X.pmf(np.arange(3, 51)))"
      ],
      "id": "0a19ad17-c482-495f-a896-512e980f285c"
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.8882712436536537"
            ]
          }
        }
      ],
      "source": [
        "S = np.arange(51)\n",
        "ind = S >= 3\n",
        "np.sum(X.pmf(S[ind]))"
      ],
      "id": "1b7a321d-81e8-42d9-aa64-d0e0ee75c00f"
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.9754620642954095"
            ]
          }
        }
      ],
      "source": [
        "# 2c\n",
        "# P[X < 10 or X > 40]\n",
        "ind = (S < 10) | (S > 40)\n",
        "np.sum(X.pmf(S[ind]))"
      ],
      "id": "34567941-b5f7-4ef2-bcf8-686014410cf2"
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.9754620642954085"
            ]
          }
        }
      ],
      "source": [
        "1 - np.sum(X.pmf(np.arange(10, 41)))"
      ],
      "id": "3975a501-61d6-45a0-a202-088f3a99971a"
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {},
      "outputs": [],
      "source": [
        "# 3\n",
        "X = sp.stats.binom(10, 0.7)"
      ],
      "id": "4e6416af-1d74-481b-8c3b-7593b341c7fa"
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.26682793200000005"
            ]
          }
        }
      ],
      "source": [
        "# 3a\n",
        "X.pmf(7)"
      ],
      "id": "c6be5fe1-003a-4556-84e3-2ac5f0e9d25b"
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.3827827864000003"
            ]
          }
        }
      ],
      "source": [
        "# 3b\n",
        "np.sum(X.pmf(np.arange(8, 11)))"
      ],
      "id": "0aebd4b8-5126-4c07-8cf1-ff3d43a87d82"
    },
    {
      "cell_type": "code",
      "execution_count": 23,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.38278278639999974"
            ]
          }
        }
      ],
      "source": [
        "1 - np.sum(X.pmf(np.arange(8)))"
      ],
      "id": "452eecca-3a42-49ee-9cd2-caa4c510a6e8"
    },
    {
      "cell_type": "code",
      "execution_count": 24,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.3827827864000003"
            ]
          }
        }
      ],
      "source": [
        "S = np.arange(11)\n",
        "ind = S >= 8\n",
        "np.sum(X.pmf(S[ind]))"
      ],
      "id": "afef57a8-75d4-47c3-bba0-69a1f2c5daca"
    },
    {
      "cell_type": "code",
      "execution_count": 25,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.38278278639999974"
            ]
          }
        }
      ],
      "source": [
        "ind = S < 8\n",
        "1 - np.sum(X.pmf(S[ind]))"
      ],
      "id": "f66506f9-9070-4aca-a41d-9a0906e62103"
    }
  ],
  "nbformat": 4,
  "nbformat_minor": 5,
  "metadata": {
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3 (ipykernel)",
      "language": "python"
    },
    "language_info": {
      "name": "python",
      "codemirror_mode": {
        "name": "ipython",
        "version": "3"
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.12.12"
    }
  }
}