{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hughes and Hase, Problem 3.5\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Introduction\n",
    "\n",
    "The probabilities in this problem can be evaluated with the cumulative distribution function, or $C_{DF}$, of a normal distribution.  (This $C_{DF}$ also has the name $\\mbox{Erf}(x)$.) \n",
    "I will use the $C_{DF}$ from the  `stats` sub-module of `scipy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean = 502\n",
    "sigma = 14"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Probability that the bag contains less than $500\\, \\mbox{g}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4432015031835318"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.cdf(500, mean, sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability is about 44%."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Probability that the bag contains at least  $530\\, \\mbox{g}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.02275013194817921"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.norm.cdf(530, mean, sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability is about 2.3%, and so, out of 1000 bags, you would expect about 23 to have a mass greater than 530 g."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Version information\n",
    "`version_information` is from J.R. Johansson (jrjohansson at gmail.com); see <a href='http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb'>Introduction to scientific computing with Python</a> . If not already installed on your machine, run `pip install --upgrade version_information` from the terminal\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.11.5 64bit [MSC v.1916 64 bit (AMD64)]"
        },
        {
         "module": "IPython",
         "version": "8.15.0"
        },
        {
         "module": "OS",
         "version": "Windows 10 10.0.26100 SP0"
        },
        {
         "module": "numpy",
         "version": "1.23.2"
        },
        {
         "module": "scipy",
         "version": "1.11.1"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.11.5 64bit [MSC v.1916 64 bit (AMD64)]</td></tr><tr><td>IPython</td><td>8.15.0</td></tr><tr><td>OS</td><td>Windows 10 10.0.26100 SP0</td></tr><tr><td>numpy</td><td>1.23.2</td></tr><tr><td>scipy</td><td>1.11.1</td></tr><tr><td colspan='2'>Sat Feb 08 13:55:56 2025 Eastern Standard Time</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.11.5 64bit [MSC v.1916 64 bit (AMD64)] \\\\ \\hline\n",
       "IPython & 8.15.0 \\\\ \\hline\n",
       "OS & Windows 10 10.0.26100 SP0 \\\\ \\hline\n",
       "numpy & 1.23.2 \\\\ \\hline\n",
       "scipy & 1.11.1 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Feb 08 13:55:56 2025 Eastern Standard Time} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.11.5 64bit [MSC v.1916 64 bit (AMD64)]\n",
       "IPython 8.15.0\n",
       "OS Windows 10 10.0.26100 SP0\n",
       "numpy 1.23.2\n",
       "scipy 1.11.1\n",
       "Sat Feb 08 13:55:56 2025 Eastern Standard Time"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%load_ext version_information\n",
    "%version_information numpy, scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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