{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "tags": [
     "remove_input"
    ]
   },
   "outputs": [],
   "source": [
    "path_data = '../../../../data/'\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('fivethirtyeight')\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Empirical Distributions\n",
    "\n",
    "In data science, the word \"empirical\" means \"observed\". Empirical distributions are distributions of observed data, such as data in random samples.\n",
    "\n",
    "In this section we will generate data and see what the empirical distribution looks like. \n",
    "\n",
    "Our setting is a simple experiment: rolling a die multiple times and keeping track of which face appears. The table `die` contains the numbers of spots on the faces of a die. All the numbers appear exactly once, as we are assuming that the die is fair."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Face</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Face\n",
       "0     1\n",
       "1     2\n",
       "2     3\n",
       "3     4\n",
       "4     5\n",
       "5     6"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die = pd.DataFrame({'Face':np.arange(1, 7, 1)})\n",
    "die"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Probability Distribution\n",
    "\n",
    "The histogram below helps us visualize the fact that every face appears with probability 1/6. We say that the histogram shows the *distribution* of probabilities over all the possible faces. Since all the bars represent the same percent chance, the distribution is called *uniform on the integers 1 through 6.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "die_bins = np.arange(0.5, 6.6, 1)\n",
    "\n",
    "unit = 'unit'\n",
    "\n",
    "fig, ax1 = plt.subplots()\n",
    "\n",
    "ax1.hist(die, bins=die_bins, density=True, alpha = 0.8, ec='white')\n",
    "\n",
    "y_vals = ax1.get_yticks()\n",
    "\n",
    "y_label = 'Percent per ' + (unit if unit else 'unit')\n",
    "\n",
    "x_label = 'Face'\n",
    "\n",
    "ax1.set_yticklabels(['{:g}'.format(x * 100) for x in y_vals])\n",
    "\n",
    "plt.ylabel(y_label)\n",
    "\n",
    "plt.xlabel(x_label)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variables whose successive values are separated by the same fixed amount, such as the values on rolls of a die (successive values separated by 1), fall into a class of variables that are called *discrete*. The histogram above is called a *discrete* histogram. Its bins are specified by the array `die_bins` and ensure that each bar is centered over the corresponding integer value. \n",
    "\n",
    "It is important to remember that the die can't show 1.3 spots, or 5.2 spots – it always shows an integer number of spots. But our visualization spreads the probability of each value over the area of a bar. While this might seem a bit arbitrary at this stage of the course, it will become important later when we overlay smooth curves over discrete histograms.\n",
    "\n",
    "Before going further, let's make sure that the numbers on the axes make sense. The probability of each face is 1/6, which is 16.67% when rounded to two decimal places. The width of each bin is 1 unit. So the height of each bar is 16.67% per unit. This agrees with the horizontal and vertical scales of the graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Empirical Distributions\n",
    "The distribution above consists of the theoretical probability of each face. It is not based on data. It can be studied and understood without any dice being rolled.\n",
    "\n",
    "*Empirical distributions,* on the other hand, are distributions of observed data. They can be visualized by *empirical histograms*. \n",
    "\n",
    "Let us get some data by simulating rolls of a die. This can be done by sampling at random with replacement from the integers 1 through 6. We have used `np.random.choice` for such simulations before. But now we will introduce a Table method for doing this. This will make it possible for us to use our familiar Table methods for visualization.\n",
    "\n",
    "The Table method is called `sample`. It draws at random with replacement from the rows of a table. Its argument is the sample size, and it returns a table consisting of the rows that were selected. An optional argument `with_replacement=False` specifies that the sample should be drawn without replacement, but that does not apply to rolling a die.\n",
    "\n",
    "Here are the results of 10 rolls of a die."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Face</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Face\n",
       "2     3\n",
       "0     1\n",
       "3     4\n",
       "2     3\n",
       "2     3\n",
       "3     4\n",
       "3     4\n",
       "4     5\n",
       "5     6\n",
       "0     1"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die.sample(10, replace=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the same method to simulate as many rolls as we like, and then draw empirical histograms of the results. Because we are going to do this repeatedly, we define a function `empirical_hist_die` that takes the sample size as its argument, rolls a die as many times as its argument, and then draws a histogram of the observed results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def empirical_hist_die(n):\n",
    "\n",
    "    unit = 'unit'\n",
    "\n",
    "    fig, ax1 = plt.subplots()\n",
    "\n",
    "    ax1.hist(die.sample(n, replace=True), bins=die_bins, density=True, alpha=0.8, ec='white')\n",
    "\n",
    "    y_vals = ax1.get_yticks()\n",
    "\n",
    "    y_label = 'Percent per ' + (unit if unit else 'unit')\n",
    "\n",
    "    x_label = 'Face'\n",
    "\n",
    "    ax1.set_yticklabels(['{:g}'.format(x * 100) for x in y_vals])\n",
    "\n",
    "    plt.ylabel(y_label)\n",
    "\n",
    "    plt.xlabel(x_label)\n",
    "\n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Empirical Histograms\n",
    "\n",
    "Here is an empirical histogram of 10 rolls. It doesn't look very much like the probability histogram above. Run the cell a few times to see how it varies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zcsvi4+OxY8cOvPnmm7KGIyIiqorkc2IbN25Eo0aNMHnyZCxYsAAtW7aEIAhIT0/HnTt34O3tjQ0bNmDDhg3G7xEEAbGxsRYJTkREJLnELl++DEEQ4OHhAQDG818ODg7w8PCAXq/HTz/9ZPI9vHeMiIgsSXKJXbp0yZI5iIiIzCb5nBgREZGtYYkREZFqKVpiJ0+exNixY9GuXTtoNBrs2rXLZHlYWBg0Go3JV//+/RVKS0REtuaxHgAsl+LiYrRv3x6hoaGYPn16hev06dMHW7ZsMU7b29tbKx4REdk4RUssKCgIQUFBAIDw8PAK13FwcIBWq7VmLCIiUgmbPyd2+vRp+Pj4wN/fH7NmzcKdO3eUjkRERDZC8pFYp06dEBUVhcGDB1e4/ODBg4iIiMCFCxdkC9e/f38EBwejefPmyMrKwvLlyxESEoIjR47AwcGhwu9JS0ur0Zj37BtDry+p0TbMZc3x7v16D2l5t6023iM1/bmoAfdRXkr8LgLW/X387eFvtX4fa/o3R6fTVblccollZWWhuLi40uXFxcXGF2XKZeTIkcb/9vX1hZ+fHzp06IBDhw4hJCSkwu+pboerU5gvwsFBX6NtmEOvL4GDQ/mHKltKA5cG0DV80mrjAb//4avpz8XWcR/lZ+3fRcD6v49169S16nhA7fubY9bHiVU9gePKlStwcXGpcaCqNG3aFO7u7khPT7foOEREpA5VHont3r0bn3zyiXF67dq12L59e7n1CgoKkJqaioEDB8qf8A/y8vJw8+ZNXuhBREQAqimx4uJi5OTkGKcLCwthMBhM1hEEAfXr18eECROwYMECswYvKioyHlUZDAZcv34dFy9ehKurK1xdXbFy5UqEhIRAq9UiKysLy5Ytg5ubG55//nmzxiEiotqpyhKbOnUqpk6dCgDo2LEjVq5cWemFHY/ju+++Q3BwsHE6KioKUVFRCA0Nxbp165CamopPP/0UhYWF0Gq16N27Nz788EOLf2xJRETqIPnCjosXL8o+eO/evVFQUFDp8oSEBNnHJCKi2sPsm51//fVXXL9+HXfv3oUoiuWW9+zZU5ZgRERE1ZFcYnfv3kVERAQ+++wzlJWVlVsuiiIEQUB+fr6sAYmIiCojucReeeUVHDhwAFOnTkXPnj2h0WgsGIuIiKh6kkvsq6++wrRp0/DWW29ZMg8REZFkkm92tre3h7e3tyWzEBERmUVyiQ0dOhSHDx+2ZBYiIiKzSC6xl19+Gbdu3cL06dORkpKCW7du4c6dO+W+iIiIrEXyOTF/f38IgoDz588jNja20vV4dSIREVmL5BJ77bXXqnwAMBERkbVJLrHIyEhL5iAiIjLbY73ZuaysDPn5+Xj48KHceYiIiCQzq8TOnTuHYcOGwd3dHT4+Pjh58iSA31+R8sILL+Do0aMWCUlERFQRySV29uxZDB48GNeuXcPYsWNNnpvYqFEjFBUV4eOPP7ZISCIioopILrE333wT3t7eOHPmDP75z3+WW967d2988803soYjIiKqiuQSO3fuHF588UU4OjpWeJVis2bNTF6gSUREZGmSS+yJJ57AE09UvnpOTg7q1asnSygiIiIpJJeYn58fDh48WOGy0tJS7N27F926dZMtGBERUXUkl9irr76KY8eOYebMmbh06RIA4NatW/jqq68QEhKCa9euYe7cuRYLSkRE9GeSb3Z+9tlnsWXLFsyfPx+7d+8GAISFhUEURTz55JPYunUrunbtarGgREREfya5xABg1KhRGDx4MJKSknD16lUYDAa0bNkS/fr1g7Ozs6UyEhERVcisEgOA+vXrY8iQIZbIQkREZBbJ58S+/PJLzJ8/v9Ll8+fPr/TCDyIiIkuQXGIbN27E/fv3K11eUlKC9evXyxKKiIhICskllpqaCj8/v0qXd+rUCZcvX5YjExERkSSSS+zhw4d48OBBpcsfPHgAvV4vSygiIiIpJJdY+/btkZiYCIPBUG6ZwWBAYmIi2rZtK2s4IiKiqkgusenTp+Pbb79FaGgozp8/D71eD71ej/Pnz2PcuHH49ttvMW3aNLMGP3nyJMaOHYt27dpBo9Fg165dJstFUURUVBTatm2LJk2aYMiQIfjxxx/NGoOIiGovyZfYjxw5EteuXUNUVBQOHz4MABAEAaIoQhAEREREYMyYMWYNXlxcjPbt2yM0NBTTp08vt3z9+vWIjo5GdHQ0dDodVq9ejeHDhyMlJQUuLi5mjUVERLWPWfeJzZs3D6NGjcL+/fuRkZEBURTRsmVLBAcHo0WLFmYPHhQUhKCgIABAeHi4yTJRFBETE4M5c+Zg6NChAICYmBjodDrExcVh4sSJZo9HRES1i6QSe/DgAV544QWMGTMGL774Il5++WVL50JmZiZycnLQt29f47x69eohMDAQZ86cYYkREZG0EqtXrx4uXLiAUaNGWTqP0aN3k7m5uZnMd3Nzw82bNyv9vrS0tBqNe8++MfT6khptw1zWHK/0t9/wdWblV5lahH1jZGcWWm24Rk6OyCu27s+Q+yg/Qx1Hq/8uAtb9ffzt4W+1fh/v/XoPaXm3H/v7dTpdlcslf5zYq1cvnDp1ChMmTHjsMI/jzy/gfHQOrjLV7XB1CvNFODhY71YBvb4EDg6OVhvvV0MdRF2w7q0Q1t7HyK4uiLpgvT+2APfREiK7ulh1PMD6+1i3Tt1av48NXBpA1/BJi21f8tWJq1atwrlz57B48WJkZGRUeKm9nLRaLQDg9m3TBs/NzS13dEZERH9Nkkusa9euyMrKQnR0NLp06YLGjRujadOmJl/u7u6yBWvevDm0Wi2SkpKM80pKSnD69Gl0795dtnGIiEi9JH+cOHz48Co/xnscRUVFSE9PB/D7DdPXr1/HxYsX4erqCk9PT4SFheHtt9+GTqeDj48P1q5dCycnJ6uemyMiItslucRiYmJkH/y7775DcHCwcToqKgpRUVEIDQ1FTEwMZs+ejQcPHmD+/PkoKCiAv78/EhISeI8YEREBeIz3icmpd+/eKCgoqHS5IAiIjIxEZGSk9UIREZFqSD4nBgBZWVmYNWsW/Pz84OnpiRMnTgAA8vLyMHfuXJw/f94SGYmIiCok+Ujsp59+wnPPPQeDwYCAgABkZWWhrKwMANCoUSOkpKRAr9dj06ZNFgtLRET0R5JLbMmSJXBxccFXX30FOzs7+Pj4mCwPCgrC559/Lnc+IiKiSkn+OPHUqVOYMmUKGjduXOFVip6enlU+SYOIiEhuZr0U08nJqdLld+/ehZ2dnSyhiIiIpDDrpZjHjx+vcJkoiti/fz/8/PzkykVERFQtySUWFhaGffv2YfXq1cjPzwfw+w3KP//8MyZNmoTvvvvOKk+3JyIiesSsl2JmZ2fjrbfewsqVK43zAMDOzg7Lly/HgAEDLJOSiIioAmbd7DxnzhyMGjUKiYmJSE9Ph8FgQMuWLRESEoLmzZtbKiMREVGFqi0xvV6PL7/8EhkZGWjYsCEGDhxY7i3MRERESqiyxHJycjB48GBcu3YNoigCAJycnLBnzx707NnTKgGJiIgqU+WFHcuXL0dGRgbCw8OxZ88eREVFwcHBAa+99pq18hEREVWqyiOxr7/+GqGhoVi+fLlxXuPGjTFlyhT88ssvaNasmcUDEhERVabKI7GcnJxyL6Ds0aMHRFHE9evXLRqMiIioOlWWWFlZGRwdHU3mPZouKSmxXCoiIiIJqr06MSMjA99++61x+t69ewCAtLQ0ODs7l1vf399fxnhERESVq7bEHr1t+c/+fHGHKIoQBMH4NA8iIiJLq7LEoqOjrZWDiIjIbFWW2Lhx46yVg4iIyGySHwBMRERka1hiRESkWiwxIiJSLZYYERGpFkuMiIhUiyVGRESqxRIjIiLVsukSi4qKgkajMflq3bq10rGIiMhGVPvYKaXpdDocOHDAOG1nZ6dgGiIisiU2X2J16tSBVqtVOgYREdkgm/44Efj9Kfrt2rVDx44dMWnSJGRkZCgdiYiIbIRNH4kFBARg8+bN0Ol0yM3NxZo1axAUFITk5GQ0bNiwwu9JS0ur0Zj37BtDr7fuu9KsOd5vD3+z+v4B3Ee5cR8th/sor3u/3kNa3u3H/n6dTlflcpsusQEDBphMBwQEwM/PD7t378bMmTMr/J7qdrg6hfkiHBz0NdqGOfT6Ejg4OFa/okzq1qlr1fEA7qMlcB8tg/sovwYuDaBr+KTFtm/zHyf+kbOzM9q2bYv09HSloxARkQ1QVYmVlJQgLS2NF3oQEREAG/84cdGiRXjuuefg4eFhPCd2//59hIaGKh2NiIhsgE2X2I0bNzBlyhTk5eXhqaeeQkBAAA4fPgwvLy+loxERkQ2w6RLbtm2b0hGIiMiGqeqcGBER0R+xxIiISLVYYkREpFosMSIiUi2WGBERqRZLjIiIVIslRkREqsUSIyIi1WKJERGRarHEiIhItVhiRESkWiwxIiJSLZYYERGpFkuMiIhUiyVGRESqxRIjIiLVYokREZFqscSIiEi1WGJERKRaLDEiIlItlhgREakWS4yIiFSLJUZERKrFEiMiItViiRERkWqxxIiISLVUUWJbt25Fx44dodVq8cwzz+DUqVNKRyIiIhtg8yWWkJCABQsWYO7cuTh27Bi6deuG0aNHIzs7W+loRESkMJsvsejoaIwbNw4TJkxAmzZtsGbNGmi1Wmzbtk3paEREpDChoKBAVDpEZUpLS9G0aVN88MEHGDZsmHH+vHnzkJqaii+//FK5cEREpDibPhLLy8tDWVkZ3NzcTOa7ubnh9u3bCqUiIiJbYdMl9oggCCbToiiWm0dERH89Nl1ijRo1gp2dXbmjrtzc3HJHZ0RE9Ndj0yVmb28PPz8/JCUlmcxPSkpC9+7dFUpFRES2oo7SAaozY8YMTJs2Df7+/ujevTu2bduGW7duYeLEiUpHIyIihdn0kRgAjBgxAlFRUVizZg169+6N5ORkxMbGwsvLS+loj+3kyZMYO3Ys2rVrB41Gg127dikdSXbr1q3Ds88+C09PT3h7e2PMmDFITU1VOpas3n//fQQGBsLT0xOenp4YMGAADh06pHQsi3n77beh0Wgwf/58paPIJioqChqNxuSrdevWSseS3a1btzB9+nR4e3tDq9Wie/fuOHHihNKxZGHzR2IAMGXKFEyZMkXpGLIpLi5G+/btERoaiunTpysdxyJOnDiByZMno0uXLhBFEStWrMCwYcNw5swZuLq6Kh1PFu7u7li6dCm8vb1hMBjwySefYPz48Thy5Aj+9re/KR1PVikpKdi+fTt8fX2VjiI7nU6HAwcOGKft7OwUTCO/goICDBw4ED169EBsbCwaNWqEzMzMWnNdgSpKrLYJCgpCUFAQACA8PFzhNJaRkJBgMr1lyxZ4eXkhOTkZgwYNUiiVvIYMGWIyvXjxYnzwwQdISUmpVSVWWFiIqVOnYuPGjVi9erXScWRXp04daLVapWNYzIYNG9CkSRNs2bLFOK9FixbKBZKZzX+cSLVDUVERDAYDNBqN0lEsoqysDPHx8SguLka3bt2UjiOrOXPmYOjQoXjmmWeUjmIRGRkZaNeuHTp27IhJkyYhIyND6Uiy+uKLL+Dv74+JEyfCx8cHvXr1wnvvvQdRtNnnXJiFR2JkFQsWLECHDh1q3R/4H374AUFBQSgpKYGTkxN27txZqz5y2759O9LT003+FV+bBAQEYPPmzdDpdMjNzcWaNWsQFBSE5ORkNGzYUOl4ssjIyMAHH3yA8PBwzJkzB5cuXUJERAQA4KWXXlI4Xc2xxMjiXn/9dSQnJ+PgwYO17nyDTqfD8ePHUVhYiMTERISFheHAgQNo37690tFqLC0tDcuWLcO///1v2NvbKx3HIgYMGGAyHRAQAD8/P+zevRszZ85UKJW8DAYDOnfujCVLlgAAOnXqhPT0dGzdupUlRlSdyMhIJCQkYP/+/bXqc/hH7O3t0apVKwBA586dce7cOWzevBmbNm1SOFnNnT17Fnl5eXj66aeN88rKynDq1Cls27YNN27cgIODg4IJ5efs7Iy2bdsiPT1d6Siy0Wq1aNOmjcm81q1b4/r16wolkhdLjCwmIiICCQkJOHDgQK28bLkiBoMBpaWlSseQxZAhQ9C5c2eTeTNmzIC3tzdeffXVWnl0VlJSgrS0NPTu3VvpKLLp0aMHrly5YjLvypUr8PT0VCiRvFhiCigqKjL+S89gMOD69eu4ePEiXF1da83/WPPmzcOePXuwc+dOaDQa5OTkAACcnJzg7OyscDp5vPHGGwgKCkKzZs1QVFSEuLg4nDhxArGxsUpHk8Wj+6b+qH79+nB1da0VH5cCwKJFi/Dcc8/Bw8PDeE7s/v37CA0NVTqabMLDwxEUFIS1a9dixIgRuHjxIt577z0sXrxY6WiysOlXsdRWx48fR3BwcLn5oaGhiImJUSCR/Cq7CjEiIgKRkZHWDWMhYWFhOH78OG7fvo0GDRrA19cXs2bNQr9+/ZSOZjFDhgxB+/btsWbNGqWjyGLSpEk4deoU8vLy8NRTTyEgIAALFy5E27ZtlY4mq0OHDmHZsmW4cuUKPDw8MHXqVEybNq1WPEidJUZERKrF+8SIiEi1WGJERKRaLDEiIlItlhgREakWS4yIiFSLJUZERKrFEiMiItViiRFZ2a5du8q9TfjR1yuvvKJ0PCJV4WOniBSyYMECtGzZ0mSej4+PQmmI1IklRqSQfv36oWvXrkrHIFI1fpxIZEPu3r2LRYsWITAwEB4eHmjWrBmef/55JCcnl1tXFEW8//776NWrF5o0aYJWrVph2LBhOHXqlMl68fHx6NevH5o2bQovLy+MGTMGly9fttYuEVkUj8SIFHLv3j3k5eWZzMvMzMS+ffswdOhQtGrVCoWFhdixYweGDh2KpKQkk6fHz549Gzt27ECfPn0wbtw4iKKIs2fP4vTp0wgMDAQAvPvuu3jjjTcQHByMsWPHori4GFu3bsXAgQNx9OjRWvmON/pr4QOAiaxs165dmDFjRoXLfv75ZzRq1MjkDdh3795F165dMXjwYGzYsAHA/96EMGHCBKxfv95kG6IoQhAEZGdno3Pnzpg7d67JmwNu3bqFbt26ISQkpFa8vJP+2ngkRqSQVatWlXvjrqurq7HASkpKcP/+fYiiCH9/f5w/f964XmJiIoDf34f1Z49er7F//348fPgQI0eONDniq1u3LgICAnDs2DG5d4nI6lhiRArp0qVLuQs7DAYD3nnnHXz00UfIzMw0Wda8eXPjf1+7dg1ubm5wc3OrdPtXr14FAHTr1q3C5fXr13/c6EQ2gyVGZEPeffddLFu2DKGhoVi0aBEaNmwIOzs7rFu3DteuXTOu9+gjw6oYDAYAQFxcHOrUKf+r/sQTvK6L1I8lRmRDEhIS0KtXr3Jv+I6KijKZbtWqFf773//izp07lR6NPboHzcPDo9a9qZjoEf5TjMiG2NnZQRRNr7U6c+YMzp49azIvJCQEALBixYpy23j0/SEhIahTpw6ioqKMR2V/lJubK1dsIsXwSIzIhgwaNAgrV67EtGnTEBgYiKtXr+Kjjz5C27ZtUVRUZFyvd+/eGDduHD788ENkZGQgKCgIAJCSkgJfX1/MnTsXLVq0wNKlS7Fw4UL0798fwcHBcHV1RXZ2Nv7zn/8gICAA77zzjlK7SiQLlhiRDXn11Vfx4MED7N27F/v27UO7du2wbds2xMfH48SJEybrbtq0Cb6+vvj444+xZMkSODs7o1OnTujZs6dxnRkzZsDHxwcbN27EunXr8PDhQzRt2hQ9evTA3//+d2vvHpHseJ8YERGpFs+JERGRarHEiIhItVhiRESkWiwxIiJSLZYYERGpFkuMiIhUiyVGRESqxRIjIiLVYokREZFqscSIiEi1/h/hJ2K8wAdMrQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "empirical_hist_die(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When the sample size increases, the empirical histogram begins to look more like the histogram of theoretical probabilities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "empirical_hist_die(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "empirical_hist_die(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we increase the number of rolls in the simulation, the area of each bar gets closer to 16.67%, which is the area of each bar in the probability histogram.\n",
    "\n",
    "## The Law of Averages\n",
    "\n",
    "What we have observed above is an instance of a general rule.\n",
    "\n",
    "If a chance experiment is repeated independently and under identical conditions, then, in the long run, the proportion of times that an event occurs gets closer and closer to the theoretical probability of the event.\n",
    "\n",
    "For example, in the long run, the proportion of times the face with four spots appears gets closer and closer to 1/6.\n",
    "\n",
    "Here \"independently and under identical conditions\" means that every repetition is performed in the same way regardless of the results of all the other repetitions."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}