{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "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": 2,
   "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": 2,
     "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": 3,
   "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": 4,
   "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>4</th>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "    </tr>\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>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</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",
       "4     5\n",
       "1     2\n",
       "0     1\n",
       "2     3\n",
       "4     5\n",
       "0     1\n",
       "1     2\n",
       "3     4\n",
       "5     6\n",
       "5     6"
      ]
     },
     "execution_count": 4,
     "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": 5,
   "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": 6,
   "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(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": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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XLFq0iEeCRETU7FlcfJs2bcKsWbOgUqmwePFi+Pv7w2Aw4NSpU0hOTkZMTAxqamqM78kjIiJqjiwuvoSEBAQHByM9PR0uLi4my6ZPn44xY8YgISGBxUdERM2axffxnTt3DpMnT25QegDg4uKCyMhInD9/XtBwREREQrO4+Pr06YMLFy40uvz8+fPo3bu3IKGIiIjEYnHxLVmyBBs2bMBXX33VYFlqaiqSk5Px1ltvCRruVrKysjBlyhQEBARAqVRi06ZNJssNBgO0Wi369OmDTp06YezYsfjzzz9Fz0VERPbB4u/4Vq5cCU9PTzzzzDOIi4tDjx49oFAokJeXh0uXLsHPzw8rVqzAihUrjNsoFAqkpKQIGriqqgqBgYHQaDSYMWNGg+XLly9HYmIiEhMToVKp8P777+Oxxx7DwYMH4ebmJmgWIiKyPxYX3/Hjx6FQKNCtWzcAMH6f5+zsjG7dukGn0+HEiRMm24hxb59arYZarQYAxMbGmiwzGAxISkrCnDlzMG7cOABAUlISVCoVvvzyS0ybNk3wPEREZF8sLr6cnBwxcwjizJkzKC4uxgMPPGCcd9ddd2H48OH4+eefWXxERGTdk1uaq+LiYgCAl5eXyXwvLy+zF+YAQG5urtX7rXDqCJ2u2urtrWHr/dXW1XKMIuAYhccxisPW+6u4WoHckotWb69SqRpd1qKK76Z/nmI1GAy3Pe1q7pd0O+WlBjg766zevql0umo4Oze8rURMjq0dbbpPjlEcHKPwOEZxtHNrB5VHe1E+2+KrOu2Bt7c3AODiRdN/JVy+fLnBUSAREclTiyq+u+++G97e3sjMzDTOq66uxk8//YShQ4dKmIyIiJoLuzvVWVlZiby8PACAXq9HYWEhjh49Cnd3d/j4+GDmzJn44IMPoFKp4O/vj/j4eLRt2xaTJk2SODkRETUHdld8hw8fRnh4uHFaq9VCq9VCo9EgKSkJs2fPxvXr1/HKK6+grKwMwcHBSEtL4z18REQEoAmnOgcMGICMjIxGl+/atQsDBgwQJJQ5YWFhKCsra/CTlJQE4MaFLfPnz8eJEydQXFyMjIwMBAYGip6LiIjsg8XFV1BQgKqqqkaXV1VVGV9OS0RE1Fw16eIWc7cEnDx5kqcTiYio2TP7Hd/mzZuxZcsW43R8fDw2bNjQYL2ysjIcO3YMo0ePFj4hERGRgMwWX1VVlfFpKABQXl4OvV5vso5CoUCbNm0wdepUxMXFiZOSiIhIIGaLb/r06Zg+fToAoH///li6dCnGjBljk2BERERisPh2hqNHj4qZg4iIyCaafB/f1atXUVhYiCtXrsBgMDRYHhoaKkgwIiIiMVhcfFeuXMG8efPw1Vdfob6+vsHymw+CLi0tFTQgERGRkCwuvhdffBHp6emYPn06QkNDoVQqRYxFREQkDouLb8+ePYiJicE777wjZh4iIiJRWXwDu5OTE/z8/MTMQkREJDqLi2/cuHHYvXu3mFmIiIhEZ3HxPf/88ygqKsKMGTNw8OBBFBUV4dKlSw1+iIiImjOLv+MLDg6GQqHAkSNHkJKS0uh6vKqTiIiaM4uL79VXXzX7kGoiIiJ7YHHxzZ8/X8wcRERENtGk1xLdVF9fj9LSUtTV1Qmdh4iISFRNKr5ff/0V48ePR5cuXeDv74+srCwAQElJCR5//HH88MMPooQkIiISisXFd+DAAYwZMwanT5/GlClTTJ7T6enpicrKSmzcuFHQcP369YNSqWzw8/jjj99y/TNnztxy/T179giai4iI7JfF3/G99dZb8PPzw3fffYeqqiokJyebLA8LC8PWrVsFDZeZmWnyXNCioiKMHDkS48ePN7tdamoq7rnnHuO0u7u7oLmIiMh+WVx8v/76KxYsWAAXFxdcu3atwfKuXbuavLRWCB06dDCZ3rhxI9zc3G5bfB4eHvD29hY0CxERtQwWn+ps1aoVWrVqfPXi4mLcddddgoS6FYPBgI0bNyIyMhJt2rQxu+5TTz0Ff39/jB49Gtu3bxctExER2R+Liy8oKAi7du265bKamhps27YNQ4YMESzYP2VmZuLMmTN46qmnGl3H1dUVb731Fj799FNs27YN//rXvzBt2jTBT8ESEZH9svhU50svvYRJkyZh1qxZmDx5MoAb37nt2bMH8fHxOH36NBITE0ULumHDBgwaNAj9+/dvdB1PT088//zzxumBAweitLQUy5cvR2RkpNnPz83NtTpbhVNH6HTVVm9vDVvvr7aulmMUAccoPI5RHLbeX8XVCuSWXLR6e5VK1egyi4vv/vvvx5o1a/DKK69g8+bNAICZM2fCYDCgffv2WLduHQYPHmx1SHMuXbqEjIwMxMfHN3nb4OBgbNq06bbrmfsl3U55qQHOzjqrt28qna4azs4uNtsfADi2drTpPjlGcXCMwuMYxdHOrR1UHu1F+WyLiw8AJk2ahDFjxiAzMxOnTp2CXq9Hjx49MGrUKLi6uooSEAA2b94MZ2dnTJgwocnb5uTk8EIXIiIyalLxAUCbNm0wduxYMbLcksFgQHJyMiZMmAA3NzeTZYsXL8ahQ4ewY8cOADcK0tHREf3790erVq2wa9curFu3Dm+++abN8hIRUfNmcfFlZGQgMzMTy5Ytu+XyV155BaNGjcLDDz8sWDgA2Lt3L06dOoW1a9c2WFZUVITTp0+bzIuPj8fZs2fh4OAAPz8/rFq16rbf7xERkXxYXHwrV65Ez549G11eXV2N5cuXC158//rXv1BWVnbLZUlJSSbTUVFRiIqKEnT/RETUslh8O8OxY8cQFBTU6PIBAwbg+PHjQmQiIiISjcXFV1dXh+vXrze6/Pr169DpbHdlIxERkTUsLr7AwEDs2LEDer2+wTK9Xo8dO3agT58+goYjIiISmsXFN2PGDBw6dAgajQZHjhyBTqeDTqfDkSNHEBUVhUOHDiEmJkbMrERERHfM4otbJk6ciNOnT0Or1WL37t0AAIVCAYPBAIVCgXnz5vHqSSIiavaadB/f3LlzMWnSJHzzzTfIz8+HwWBAjx49EB4eju7du4sUkYiISDgWFd/169fx+OOPIzIyEk8++aTJ8zCJiIjsiUXf8d1111347bffTF4KS0REZI8svrhlxIgR2L9/v5hZiIiIRGdx8b333nv49ddfsXDhQuTn59/ytgYiIqLmzuKLWwYPHgyDwYDExEQkJiaiVatWcHR0NFlHoVDg/PnzgockIiISisXF99hjj0GhUIiZhYiISHQWF98/HwhNRERkjyz+jo+IiKglaFLxFRQU4IUXXkBQUBB8fHywb98+AEBJSQlefvllHDlyRIyMREREgrH4VOeJEyfw8MMPQ6/XIyQkBAUFBcb7+jw9PXHw4EHodDqsWrVKtLBERER3yuLiW7RoEdzc3LBnzx44ODjA39/fZLlarcbXX38tdD4iIiJBWXyqc//+/YiOjkbHjh1veXWnj48PLly4IGg4IiIioTXpRbRt27ZtdPmVK1fg4OAgSCgiIiKxNOlFtHv37r3lMoPBgG+++QZBQUFC5bojWq0WSqXS5KdXr15SxyIiombA4u/4Zs6ciejoaLz//vuYMGECgBtvXv/rr7+g1Wpx+PBhbN26VbSgTaVSqZCenm6c5tEoEREBTXwR7dmzZ/HOO+9g6dKlxnnAjVJ5++238dBDD4mT0gqtW7eGt7e31DGIiKiZadKLaOfMmYNJkyZhx44dyMvLg16vR48ePRAREYG7775brIxWyc/PR0BAABwdHRESEoI33niDL8slIqLbF59Op0NGRgby8/Ph4eGB0aNHIzY21hbZrBYSEoLVq1dDpVLh8uXLWLZsGdRqNbKzs+Hh4XHLbXJzc63eX4VTR+h01VZvbw1b76+2rpZjFAHHKDyOURy23l/F1Qrklly0enuVStXoMrPFV1xcjDFjxuD06dMwGAwAgLZt22Lr1q0IDQ21OpDY/nnKNSQkBEFBQdi8eTNmzZp1y23M/ZJup7zUAGdnndXbN5VOVw1nZxeb7Q8AHFs72nSfHKM4OEbhcYziaOfWDiqP9qJ8ttmrOt9++23k5+cjNjYWW7duhVarhbOzM1599VVRwojF1dUVffr0QV5entRRiIhIYmaP+P7v//4PGo0Gb7/9tnFex44dER0djXPnzqFr166iBxRCdXU1cnNzERYWJnUUIiKSmNkjvuLiYgwdOtRk3rBhw2AwGFBYWChqsDuxYMEC7Nu3D/n5+fjll18wdepUXLt2DRqNRupoREQkMbNHfPX19XBxMT2ve3O6utq2X3Q2xfnz5xEdHY2SkhJ06NABISEh2L17N3x9faWORkREErvtVZ35+fk4dOiQcbqiogLAjasgXV1dG6wfHBwsYDzrrF+/XuoIRETUTN22+LRaLbRabYP5/7zAxWAwQKFQoLS0VLh0REREAjNbfImJibbKQUREZBNmiy8qKspWOYiIiGzC4rczEBERtQQsPiIikhUWHxERyQqLj4iIZIXFR0REssLiIyIiWWHxERGRrLD4iIhIVlh8REQkKyw+IiKSFRYfERHJCouPiIhkhcVHRESywuIjIiJZYfEREZGsNOviS0hIwP333w8fHx/4+fkhMjISx44dM7vNmTNnoFQqG/zs2bPHRqmJiKg5M/siWqnt27cPzzzzDAYNGgSDwYB3330X48ePx88//wx3d3ez26ampuKee+4xTt9ufSIikodmXXxpaWkm02vWrIGvry+ys7PxyCOPmN3Ww8MD3t7eYsYjIiI71KxPdf5TZWUl9Ho9lErlbdd96qmn4O/vj9GjR2P79u3ihyMiIrvQrI/4/ikuLg79+vXDkCFDGl3H1dUVb731FoYNG4bWrVsjIyMD06ZNQ1JSEiIjIxvdLjc31+pcFU4dodNVW729NWy9v9q6Wo5RBByj8DhGcdh6fxVXK5BbctHq7VUqVaPL7Kb4XnvtNWRnZ2PXrl1wcHBodD1PT088//zzxumBAweitLQUy5cvN1t85n5Jt1NeaoCzs87q7ZtKp6uGs7OLzfYHAI6tHW26T45RHByj8DhGcbRzaweVR3tRPtsuTnXOnz8fqamp2LFjB7p3797k7YODg5GXlyd8MCIisjvN/ohv3rx5SEtLQ3p6Onr16mXVZ+Tk5PBCFyIiAtDMi2/u3LnYunUrPv/8cyiVShQXFwMA2rZtC1dXVwDA4sWLcejQIezYsQMAsHnzZjg6OqJ///5o1aoVdu3ahXXr1uHNN9+UahhERNSMNOviW7duHQBg3LhxJvPnzZuH+fPnAwCKiopw+vRpk+Xx8fE4e/YsHBwc4Ofnh1WrVpn9fo+IiOSjWRdfWVnZbddJSkoymY6KikJUVJRIiYiIyN7ZxcUtREREQmHxERGRrLD4iIhIVlh8REQkKyw+IiKSFRYfERHJCouPiIhkhcVHRESywuIjIiJZYfEREZGssPiIiEhWWHxERCQrLD4iIpIVFh8REckKi4+IiGSFxUdERLLC4iMiIllh8RERkay02OJbt24d+vfvD29vb9x3333Yv3+/1JGIiKgZaJHFl5aWhri4OLz88sv48ccfMWTIEEyePBlnz56VOhoREUmsRRZfYmIioqKiMHXqVPTu3RvLli2Dt7c31q9fL3U0IiKSmKKsrMwgdQgh1dTUoHPnzvjkk08wfvx44/y5c+fi2LFjyMjIkC4cERFJrsUd8ZWUlKC+vh5eXl4m8728vHDx4kWJUhERUXPR4orvJoVCYTJtMBgazCMiIvlpccXn6ekJBweHBkd3ly9fbnAUSERE8tPiis/JyQlBQUHIzMw0mZ+ZmYmhQ4dKlIqIiJqL1lIHEMNzzz2HmJgYBAcHY+jQoVi/fj2Kioowbdo0qaMREZHEWtwRHwBMmDABWq0Wy5YtQ1hYGLKzs5GSkgJfX1+po92RrKwsTJkyBQEBAVAqldi0aZPUkQSVkJCA+++/Hz4+PvDz80NkZCSOHTsmdSxBffzxxxg+fDh8fHzg4+ODhx56CN9++63UsUTzwQcfQKlU4pVXXpE6iqC0Wi2USqXJT69evaSOJbiioiLMmDEDfn5+8Pb2xtChQ7Fv3z6pY92xFnnEBwDR0dGIjo6WOoagqqqqEBgYCI1GgxkzZkgdR3D79u3DM888g0GDBsFgMODdd9/F+PHj8fPPP8Pd3V3qeILo0qULFi9eDD8/P+j1emzZsgVPPPEEvv/+e9xzzz1SxxPUwYMHsWHDBvTt21fqKKJQqVRIT083Tjs4OEiYRnhlZWUYPXo0hg0bhpSUFHh6euLMmTMt4lqJFlt8LZFarYZarQYAxMbGSpxGeGlpaSbTa9asga+vL7Kzs/HII49IlEpYY8eONZleuHAhPvnkExw8eLBFFV95eTmmT5+OlStX4v3335c6jihat24Nb29vqWOIZsWKFejUqRPWrFljnNe9e3fpAgmoRZ7qpJahsrISer0eSqVS6iiiqK+vR2pqKqqqqjBkyBCp4whqzpw5GDduHO677z6po4gmPz8fAQEB6N+/P55++mnk5+dLHUlQO3fuRHBwMKZNmwZ/f3+MGDECa9euhcFg/8884REfNVtxcXHo169fiyuFP/74A2q1GtXV1Wjbti0+//zzFnU6cMOGDcjLyzM5UmhpQkJCsHr1aqhUKly+fBnLli2DWq1GdnY2PDw8pI4niPz8fHzyySeIjY3FnDlzkJOTg3nz5gEAnn32WYnT3RkWHzVLr732GrKzs7Fr164W992JSqXC3r17UV5ejh07dmDmzJlIT09HYGCg1NHuWG5uLpYsWYL//Oc/cHJykjqOaB566CGT6ZCQEAQFBWHz5s2YNWuWRKmEpdfrMXDgQCxatAgAMGDAAOTl5WHdunUsPiKhzZ8/H2lpafjmm29azHcKf+fk5ISePXsCAAYOHIhff/0Vq1evxqpVqyROducOHDiAkpIS3HvvvcZ59fX12L9/P9avX4/z58/D2dlZwoTicHV1RZ8+fZCXlyd1FMF4e3ujd+/eJvN69eqFwsJCiRIJh8VHzcq8efOQlpaG9PT0Fnl5+K3o9XrU1NRIHUMQY8eOxcCBA03mPffcc/Dz88NLL73UYo8Cq6urkZubi7CwMKmjCGbYsGE4efKkybyTJ0/Cx8dHokTCYfHZkcrKSuO/KPV6PQoLC3H06FG4u7u3iD+Mc+fOxdatW/H5559DqVSiuLgYANC2bVu4urpKnE4Yb775JtRqNbp27YrKykp8+eWX2LdvH1JSUqSOJoib97T9XZs2beDu7t4iTuXetGDBAjz88MPo1q2b8Tu+a9euQaPRSB1NMLGxsVCr1YiPj8eECRNw9OhRrF27FgsXLpQ62h1rca8lasn27t2L8PDwBvM1Gg2SkpIkSCSsxq7enDdvHubPn2/bMCKZOXMm9u7di4sXL6Jdu3bo27cvXnjhBYwaNUrqaKIZO3YsAgMDsWzZMqmjCObpp5/G/v37UVJSgg4dOiAkJASvv/46+vTpI3U0QX377bdYsmQJTp48iW7dumH69OmIiYmx+wf+s/iIiEhWeB8fERHJCouPiIhkhcVHRESywuIjIiJZYfEREZGssPiIiEhWWHxERCQrLD4iO7Bp06YGb/y++fPiiy9KHY/IrvCRZUR2JC4uDj169DCZ5+/vL1EaIvvE4iOyI6NGjcLgwYOljkFk13iqk8jOXblyBQsWLMDw4cPRrVs3dO3aFY8++iiys7MbrGswGPDxxx9jxIgR6NSpE3r27Inx48dj//79JuulpqZi1KhR6Ny5M3x9fREZGYnjx4/bakhEouIRH5EdqaioQElJicm8M2fOYPv27Rg3bhx69uyJ8vJyJCcnY9y4ccjMzDR5K8Ls2bORnJyMkSNHIioqCgaDAQcOHMBPP/2E4cOHAwA++ugjvPnmmwgPD8eUKVNQVVWFdevWYfTo0fjhhx9a5DsSSV74kGoiO7Bp0yY899xzt1z2119/wdPT0+RN9VeuXMHgwYMxZswYrFixAsD/3u4xdepULF++3OQzDAYDFAoFzp49i4EDB+Lll182eSNGUVERhgwZgoiIiBbxwlySNx7xEdmR9957r8Fbsd3d3Y2lV11djWvXrsFgMCA4OBhHjhwxrrdjxw4AN94l9083XzPzzTffoK6uDhMnTjQ5snR0dERISAh+/PFHoYdEZHMsPiI7MmjQoAYXt+j1enz44Yf47LPPcObMGZNld999t/G/T58+DS8vL3h5eTX6+adOnQIADBky5JbL27RpY210omaDxUdk5z766CMsWbIEGo0GCxYsgIeHBxwcHJCQkIDTp08b17t5OtMcvV4PAPjyyy/RunXDvx5ateL1cGT/WHxEdi4tLQ0jRoxAUlKSyXytVmsy3bNnT3z33Xe4dOlSo0d9N+8R7NatW4t7mzjRTfznG5Gdc3BwgMFgeo3azz//jAMHDpjMi4iIAAC8++67DT7j5vYRERFo3bo1tFqt8ejv7y5fvixUbCLJ8IiPyM498sgjWLp0KWJiYjB8+HCcOnUKn332Gfr06YPKykrjemFhYYiKisKnn36K/Px8qNVqAMDBgwfRt29fvPzyy+jevTsWL16M119/HQ8++CDCw8Ph7u6Os2fP4r///S9CQkLw4YcfSjVUIkGw+Ijs3EsvvYTr169j27Zt2L59OwICArB+/XqkpqZi3759JuuuWrUKffv2xcaNG7Fo0SK4urpiwIABCA0NNa7z3HPPwd/fHytXrkRCQgLq6urQuXNnDBs2DE899ZSth0ckON7HR0REssLv+IiISFZYfEREJCssPiIikhUWHxERyQqLj4iIZIXFR0REssLiIyIiWWHxERGRrLD4iIhIVlh8REQkK/8PVxcf9nOU5ggAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "empirical_hist_die(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "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": "3.6.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}