{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Global constant\n",
    "YEARS_END = 26"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The switch probability matrix `P`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = np.array([[0.70, 0.15, 0.15],\n",
    "              [0.20, 0.80, 0.15],\n",
    "              [0.10, 0.05, 0.70]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The proportion `X` of customers currently in each store"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array([[0.15], \n",
    "              [0.20], \n",
    "              [0.65]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial:\n",
      "[[0.15]\n",
      " [0.2 ]\n",
      " [0.65]]\n"
     ]
    }
   ],
   "source": [
    "%precision 4\n",
    "\n",
    "print('Initial:')\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initially ...\n",
      "\n",
      "Customers in Store A: 15.00%\n",
      "Customers in Store B: 20.00%\n",
      "Customers in Store C: 65.00%\n"
     ]
    }
   ],
   "source": [
    "print('Initially ...')\n",
    "print()\n",
    "print(f'Customers in Store A: {X[0][0]:.2%}')\n",
    "print(f'Customers in Store B: {X[1][0]:.2%}')\n",
    "print(f'Customers in Store C: {X[2][0]:.2%}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate the Markov chain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "After year 1:\n",
      "[[0.2325]\n",
      " [0.2875]\n",
      " [0.48  ]]\n",
      "\n",
      "After year 2:\n",
      "[[0.2779]\n",
      " [0.3485]\n",
      " [0.3736]]\n",
      "\n",
      "After year 3:\n",
      "[[0.3028]\n",
      " [0.3904]\n",
      " [0.3067]]\n",
      "\n",
      "After year 4:\n",
      "[[0.3166]\n",
      " [0.4189]\n",
      " [0.2645]]\n",
      "\n",
      "After year 5:\n",
      "[[0.3241]\n",
      " [0.4381]\n",
      " [0.2378]]\n",
      "\n",
      "After year 6:\n",
      "[[0.3283]\n",
      " [0.451 ]\n",
      " [0.2208]]\n",
      "\n",
      "After year 7:\n",
      "[[0.3305]\n",
      " [0.4596]\n",
      " [0.2099]]\n",
      "\n",
      "After year 8:\n",
      "[[0.3318]\n",
      " [0.4652]\n",
      " [0.203 ]]\n",
      "\n",
      "After year 9:\n",
      "[[0.3325]\n",
      " [0.469 ]\n",
      " [0.1985]]\n",
      "\n",
      "After year 10:\n",
      "[[0.3329]\n",
      " [0.4715]\n",
      " [0.1957]]\n",
      "\n",
      "After year 11:\n",
      "[[0.3331]\n",
      " [0.4731]\n",
      " [0.1938]]\n",
      "\n",
      "After year 12:\n",
      "[[0.3332]\n",
      " [0.4742]\n",
      " [0.1926]]\n",
      "\n",
      "After year 13:\n",
      "[[0.3333]\n",
      " [0.4749]\n",
      " [0.1919]]\n",
      "\n",
      "After year 14:\n",
      "[[0.3333]\n",
      " [0.4753]\n",
      " [0.1914]]\n",
      "\n",
      "After year 15:\n",
      "[[0.3333]\n",
      " [0.4756]\n",
      " [0.1911]]\n",
      "\n",
      "After year 16:\n",
      "[[0.3333]\n",
      " [0.4758]\n",
      " [0.1909]]\n",
      "\n",
      "After year 17:\n",
      "[[0.3333]\n",
      " [0.476 ]\n",
      " [0.1907]]\n",
      "\n",
      "After year 18:\n",
      "[[0.3333]\n",
      " [0.476 ]\n",
      " [0.1906]]\n",
      "\n",
      "After year 19:\n",
      "[[0.3333]\n",
      " [0.4761]\n",
      " [0.1906]]\n",
      "\n",
      "After year 20:\n",
      "[[0.3333]\n",
      " [0.4761]\n",
      " [0.1905]]\n",
      "\n",
      "After year 21:\n",
      "[[0.3333]\n",
      " [0.4761]\n",
      " [0.1905]]\n",
      "\n",
      "After year 22:\n",
      "[[0.3333]\n",
      " [0.4762]\n",
      " [0.1905]]\n",
      "\n",
      "After year 23:\n",
      "[[0.3333]\n",
      " [0.4762]\n",
      " [0.1905]]\n",
      "\n",
      "After year 24:\n",
      "[[0.3333]\n",
      " [0.4762]\n",
      " [0.1905]]\n",
      "\n",
      "After year 25:\n",
      "[[0.3333]\n",
      " [0.4762]\n",
      " [0.1905]]\n"
     ]
    }
   ],
   "source": [
    "for year in range(1, YEARS_END):\n",
    "    X = P@X\n",
    "    print()\n",
    "    print(f'After year {year}:')\n",
    "    print(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## At the end ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "After 25 years ...\n",
      "\n",
      "Customers in Store A: 33.33%\n",
      "Customers in Store B: 47.62%\n",
      "Customers in Store C: 19.05%\n"
     ]
    }
   ],
   "source": [
    "print(f'After {YEARS_END - 1} years ...')\n",
    "print()\n",
    "print(f'Customers in Store A: {X[0][0]:.2%}')\n",
    "print(f'Customers in Store B: {X[1][0]:.2%}')\n",
    "print(f'Customers in Store C: {X[2][0]:.2%}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## After about 19 years, the percentages of customers in the stores reached a steady state (to 4 decimal places)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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