diff --git a/Coin Flip homework - dean.ipynb b/Coin Flip homework - dean.ipynb
new file mode 100644
index 0000000..fdcb1d5
--- /dev/null
+++ b/Coin Flip homework - dean.ipynb	
@@ -0,0 +1,360 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:11f10adbde253c12f3ba3ca5dff476ab168d924a1e79289e9b37ccbeaf69d208"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import random\n",
+      "import statistics as st\n",
+      "import matplotlib.pyplot as plt\n",
+      "import math\n",
+      "import seaborn as sb\n",
+      "from operator import sub\n",
+      "import numpy as np"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sides = [0, 1]\n",
+      "\n",
+      "def coin_flip():\n",
+      "    return random.choice(sides)\n",
+      "\n",
+      "coin_flip()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 4,
+       "text": [
+        "0"
+       ]
+      }
+     ],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\"\"\"Generate list of total number of head results at x intervals.\"\"\"\n",
+      "\n",
+      "def flip_counts(n):\n",
+      "    tails = []\n",
+      "    bigger_list = []\n",
+      "    xs = [pow(2, exp) for exp in range(n) if pow(2, exp) < n]\n",
+      "    xs.append(n)\n",
+      "    counter = 0\n",
+      "    while counter <= n:\n",
+      "        coin_flip()\n",
+      "        if coin_flip() == 1:  \n",
+      "            tails.append(1)\n",
+      "        if counter in xs:\n",
+      "            temp_list = tails[:]\n",
+      "            bigger_list.append(temp_list)\n",
+      "        counter += 1\n",
+      "\n",
+      "    ys = [len(i) for i in bigger_list]\n",
+      "    \n",
+      "    return xs, ys\n",
+      "\n",
+      "\n",
+      "\n",
+      "flip_counts(100000)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 5,
+       "text": [
+        "([1,\n",
+        "  2,\n",
+        "  4,\n",
+        "  8,\n",
+        "  16,\n",
+        "  32,\n",
+        "  64,\n",
+        "  128,\n",
+        "  256,\n",
+        "  512,\n",
+        "  1024,\n",
+        "  2048,\n",
+        "  4096,\n",
+        "  8192,\n",
+        "  16384,\n",
+        "  32768,\n",
+        "  65536,\n",
+        "  100000],\n",
+        " [2,\n",
+        "  2,\n",
+        "  3,\n",
+        "  5,\n",
+        "  9,\n",
+        "  18,\n",
+        "  32,\n",
+        "  60,\n",
+        "  123,\n",
+        "  238,\n",
+        "  516,\n",
+        "  1024,\n",
+        "  2038,\n",
+        "  4107,\n",
+        "  8235,\n",
+        "  16375,\n",
+        "  32783,\n",
+        "  50023])"
+       ]
+      }
+     ],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\n",
+      "x1 = np.array([1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 100000])\n",
+      "ys = np.array([1, 1, 2, 5, 7, 15, 29, 63, 128, 241, 500, 1016, 2057, 4086, 8259, 16414, 32728, 49956])\n",
+      "y1 = x1 - ys\n",
+      "\n",
+      "\n",
+      "plt.plot(y1)\n",
+      "plt.xticks(range(19), x1)\n",
+      "plt.xlabel(\"Powers of 2\")\n",
+      "plt.ylabel(\"Difference\")\n",
+      "plt.title(\"Heads vs Tails\")\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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qxZh6sy7F1Pd1DYaYFix+maWrN3H8EQ286fQp+9y3UarrV0h9nTieMrPZ7v4o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\nosQhIiJ5+f+TkGWeGSitmgAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x108ceea90>"
+       ]
+      }
+     ],
+     "prompt_number": 13
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x1 = np.array([1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 100000])\n",
+      "ys = np.array([1, 1, 2, 5, 7, 15, 29, 63, 128, 241, 500, 1016, 2057, 4086, 8259, 16414, 32728, 49956])\n",
+      "y1 = x1 / ys\n",
+      "\n",
+      "plt.plot(y1)\n",
+      "plt.xticks(range(19), x1)\n",
+      "plt.xlabel(\"Powers of 2\")\n",
+      "plt.ylabel(\"Ratio\")\n",
+      "plt.title(\"Heads vs Tails\")\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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N5raDoRD1gXamjE1uMao3c2dU4MPdD/Cxo6fGvd6WmhZyc/yMTtHwFcaYzDOi\nu5cEWjroDoZS3gU0UklhLjMmlrKmOkBLW2dc63QHg3xY38KEyiL8Kbp72RiTeeIqAYjIQcAJQBbw\ngqouTWamEiU8CNxQdAGNNG9GJWurA6xcX89RB47t8/07Glrp6g7ZDWDGmKTqswQgIhcDjwDTgWnA\nwyJyeZLzlRB7bgIb4gDgTRb/bpyTxdskMMaYVIinBPBN4ChVrQMQkZuAF4E7k5mxREjlTGC9mTK2\nmLLiXJavqyMYDOH3916tY9NAGmNSIZ42AH/45A+gqrW4LpzDXjgAVJYNXRsAeJPEzKikubWT9R/2\nfb/b7jGA7CYwY0wSxVMCWCYiP8Nd8fuAy4F3k5qrBBkubQDgqoFeXraNZWvqmDmh92GPqmtbKMjL\nojzFI5gaYzJLPCWALwIdwF3A3d7jK5OZqUSpC7SRk+2npCBnqLPC7GnlZPl9fU4W39kVZHt9KxNG\nF6Vs/gJjTGbqswSgqruAhSnIS8LVB9qoKMkbFifSgrxsZk0exaqNDexsbmdUceyr++31uwiGQtYA\nbIxJuh4DgIi8o6qHikgwxsshVc1KYr4GraOzm6ZdnUwaRnfSHjyzklUbG1i+to6PHjwh5nu21Fr9\nvzEmNXoMAKp6qPd/n2oiERn2ldMNTUM3CmhP5u03mgeeW8OyXgLAVusBZIxJkXjuA3g96nkW8FbS\ncpQge7qADp9YNba8gDGjCli5oZ6u7lgFq4h7AIZRycUYMzL1VgX0PHC89zjybNUNPJrkfA1a3TC5\nCSySz+dj3sxK/rFkCx9s3smB0yr2eU91bQvFBTmUFg59w7UxZmTrrQroRAAR+T9V/Y/UZSkxdncB\nHWaDqc3bzwWAd9fW7RMA2ju7qWloZdbkUcOi4doYM7LFcx/AQhH5BFCMuw8gC5iuqtcnNWeDtLsK\naJj1pZfJ5eTm+Fm+ro7PnLT/Xq9tq2shBEyw+n9jTArEEwAW4+bt3R94CTiONKgCGsq5gHuTk+1n\n9tQKlq6pZcfOVsaM2jPhe7j+f5L1ADLGpEA8N4IJ8G/Aw8BPgKOAKcnMVCLUBdopLsghL2f49Vad\nt583Scya2r2Wh8cAmmABwBiTAvEEgO2qGgJWA/NUdSswLrnZGpxQKERDoG1YNQBH6mmWMJsFzBiT\nSvFUAa0UkZ8Dvwb+ICITgOFVsR6lubWTjq7gsOoCGqmiNJ9JVcWs3riT9o5u8nJdKaW6ppmyolyK\nh8HQFcbIhHYyAAAX+ElEQVSYka/XEoCICHAj8GdVfQ+4ATcnwI6k52wQwj2AhmsJAODg/Srp6g6y\namMDALvaOqkLtNsNYMaYlOkxAIjIjcASQIF8EckGZgMXALHvYhomhss8AL2ZN3PvaqBN25sAq/83\nxqROb1VAl+J6/kwAvg98CxgLfEpVn05B3gZsON4FHG3GhFKK8rNZtraWUGgWG7e5ADCcxi4yxoxs\nvVUBBVR1m6ouAY4ElgGHDPeTPwyfqSB7k+X3M3dGJfWBdqprWti03U0UY4PAGWNSpbcAEFnNUwt8\nQ1XTZCaw4TMRTG8iq4E2bbMqIGNMasXTCwigzesKmhYaAm1k+X2UFecOdVZ6NXd6BT7c/QC1gTYq\nS/MoyIv3KzHGmMHp7WwzR0TWe48nRDwGNx/AjHg2ICLzgR+FxxaKWH4kcDNueIlq4BJV7Yg/6z2r\nC7RRXpKHf5iPp1NSmMuMiaV8UN1IKAQHefcHGGNMKvQWAGYNNnERWQh8FmiOWu4D7gDOU9V1IvJF\nYDqux9GgdHUHaWzuYNbkUYNNKiXmzRzN2mqv/t+6gBpjUqi30UA3JCD9NcC5wO+ils8C6oBrRGQu\n8LiqDvrkD24imBDDv/4/bN6MSh5+aR1gDcDGmNSKZyiIAVPVxUBXjJdGAwuAnwMnAyeJyIkx3tdv\nu3sAlQ3fLqCRpowtZpTXVmElAGNMKg1Vi2MdsCZ81S8iTwFHAM/3tlJVVUmfCS/fuBOAKRNG9fr+\neNKK12DTOvPYGbyytJqDDxhHboIGr0vU/g2nzykZaVmeUp+W5Wlo0oplqALAOqBYRGaq6lrgo8Bv\n+1qppqapz4Q3VLsAkOvr+f1VVSVxpRWPRKR10iET+MwpMqzylMh0hmtalqfUp2V5Sn1avQWRVAWA\nEICIXAAUq+oiEbkc+KPXIPyqqj6ZiA3tuQksPaqAjDFmqCQ9AHiNyQu8x/dHLH8emJ/o7dWnyU1g\nxhgz1JLaCDwU6gNtFOZl2w1VxhjThxEVAEKhELWBNrv6N8aYOIyoANDa3kV7R7fV/xtjTBxGVABI\nl0HgjDFmOBhhAWD4zwNgjDHDxYgKAOkwD4AxxgwXIywAWBWQMcbEa4QFACsBGGNMvEZUAKgLtOHz\nwaiS4T0RjDHGDAcjKgDUexPBZPlH1G4ZY0xSjJgzZXcwSENTBxUlVv1jjDHxGDEBoLG5g2AoZF1A\njTEmTiMmANRZA7AxxvTLiAsA1gXUGGPiM2ICQIN3D4CVAIwxJj4jJgDYMBDGGNM/IyYAhO8Criyz\nEoAxxsRjxASAukAbeblZFNpEMMYYE5cREwDqA21UlOTh8/mGOivGGJMWRkQAaG3voqWtyxqAjTGm\nH0ZEAKhvslFAjTGmv0ZEAGjYfROY9QAyxph4jYgAYDeBGWNM/42QAGA3gRljTH8lPQCIyHwReb6X\n1+8QkR8OZhvhiWAq7B4AY4yJW1IDgIgsBBYBMSvnReTLwFwgNJjthANAebG1ARhjTLySXQJYA5wL\n7NM5X0QWAEcBt8d6vT/qAm2UFeWSkz0iarSMMSYlknrGVNXFQFf0chEZD1wPfJVBnvyDoRD1gXZr\nADbGmH4aqnETPgmMBp4AxgGFIrJKVe/rbaWqqpJ9ljUE2ugOhphQVRzz9f6kNVCJSsvylPq0LE+p\nT8vyNDRpxTIkAUBVfw78HEBELgUO6OvkD1BT07TPsnVbAwAU5WXFfD2WqqqSuN+bqrQsT6lPy/KU\n+rQsT6lPq7cgkqpK8xCAiFwgIl/s6fWBqLeZwIwxZkCSXgJQ1Q3AAu/x/TFev3cw6dtNYMYYMzBp\n323GJoIxxpiBSfsAUG93ARtjzICkfQCoC7SRneWnpDBnqLNijDFpJe0DQH2gjcpSmwjGGGP6K60D\nQEdnN027Oq0B2BhjBiCtA0BDk9X/G2PMQKV1ALAeQMYYM3AjIgBYCcAYY/ovrQNAuAuotQEYY0z/\npXUAsCogY4wZuLQOAPU2DIQxxgxYmgeAdooLcsjLyRrqrBhjTNpJ2wAQCoW8m8Ds6t8YYwYibQNA\nc2snHV1Bq/83xpgBStsAYIPAGWPM4KRtALB5AIwxZnBGQACwKiBjjBmItA0ANhWkMcYMTtoGgDq7\nC9gYYwYlbQNAQ6CNLL+PsuLcoc6KMcakpbQNAHWBNspL8vDbRDDGGDMgaRkAurqDNDZ3WP2/McYM\nQloGgIamdkJYDyBjjBmMtAwANgicMcYMXnayNyAi84EfqeqJUcsvAK4GuoDlwJWqGoonTZsIxhhj\nBi+pJQARWQgsAvKilhcA3wdOUNVjgTLgrHjTtS6gxhgzeMmuAloDnAtEd9VpA45R1TbveTbQGm+i\nDbtLANYGYIwxA5XUAKCqi3FVPNHLQ6paAyAiXwOKVPUf8aZrJQBjjBm8pLcB9ERE/MCPgf2A8+JZ\np6qqBIDGXR0UFeQwZVL5gLcfTisREpWW5Sn1aVmeUp+W5Wlo0oplyAIAcDuuKugT8Tb+1tQ0EQqF\n2F6/i6qyAmpqmga04aqqkgGvm6y0LE+pT8vylPq0LE+pT6u3IJKqABCC3T1/ioG3gMuAl4DnRATg\nNlV9pK+EWtu7aO/otnsAjDFmkJIeAFR1A7DAe3x/xEsDmsi3ziaCMcaYhEi7G8FsHgBjjEmMtAsA\nDXYTmDHGJETaBQDrAmqMMYmRdgHAZgIzxpjESLsAUBdow+eDUSU2EYwxxgxG2gWA+kAbo4rzyPKn\nXdaNMWZYSauzaHcwSEOTTQRjjDGJkFYBoLG5g2AoZF1AjTEmAdIqANg8AMYYkzhpFQDqrQuoMcYk\nTJoFACsBGGNMoqRVALBhIIwxJnHSKgBYFZAxxiROWgWAukAbeTlZFOUP5TQGxhgzMqRVAKgPtFFR\nmofPFz3FsDHGmP5KmwDQ2t5FS1uXNQAbY0yCpE0AqN3ZClj9vzHGJEraBICaBhcAKq0HkDHGJET6\nBICduwArARhjTKKkTwDYXQKwAGCMMYmQPgFgdxuAVQEZY0wipE8A8EoA5SVWAjDGmERInwCwcxdl\nRbnkZKdNlo0xZlhLm7Np7c42awA2xpgESnoAEJH5IvJ8jOVni8gbIvKaiHyhr3S6uoPWBdQYYxIo\nqQFARBYCi4C8qOU5wC3AKcDxwJdEZExf6VkJwBhjEifZJYA1wLlA9OA9BwJrVLVRVTuBV4Dj+krM\nuoAaY0ziJDUAqOpioCvGS6VAY8TzJqCsr/SsC6gxxiTOUI2r3AiURDwvARp6W+Gxm89J6BCgVVUl\nfb8pxWlZnlKfluUp9WlZnoYmrViGKgCsBvYXkXKgBVf985MhyosxxmSkVAWAEICIXAAUq+oiEbkG\neBpXDXWnqm5LUV6MMcYAvlAoNNR5MMYYMwTS5kYwY4wxiWUBwBhjMpQFAGOMyVBD1Quo30RkPvAj\nVT1xEGnkAHcBU3F3J9+kqo8NIr0xwBLgJFV9f4Bp+IHfArOAIPBFVdV+prH7s/HytAgYhbsB7xJV\n3RBHGlneerNwjfZXADnA/wHdQLuX1o4483QtcLaXxi9U9V5v+YXAV1V1QT/26ZBY+RCRrwCXefn9\ngao+EpXGPt83sAX4GxD+vn6lqg+KyBnA9d6yN1X1P2Lk6W323L+yXlUv85bfCqxW1du95/8JnO+9\n7wlV/V6MfdoPuAf3na8ArlLVUE/reusfAPwTeByYHLFPq2Kl5a3j997/iKreLiKFwP2446MD+Kz3\n+S0BTvLSiJWva7z3tgE/V9X7vWPmFuBwQIDtQD0Rx3CMzyZWOmXA74EJwH7AacA6YhzHInIVcKmX\n55963130PtUAU9hzHF/hfaYr8Y4fYDzws4iv92jgHOAfwK3ePk3F3aPUguuw8u/efm4D1rPn2LkN\n+Ij3Xrz3BYE/et/TNGAuUAg8AMwEdgHLgQuATwJfwt0zdZOqPi4iBd5nUuWle6mq1orI0V6+u4Bn\nIo6tG4CPecu/rqpvEoe0KAH0NKTEAFwE1KjqccDpwC8Gkacc4HbcwTEYpwJFqnos8D3gf/qZj+jP\n5sfA71T1eNwJbW6cSZ0FBL18/DfwA9yB9lUv6C4GvhVnnk4AjvFO8icAM7zlh+JO2P3dp33yISJF\nwH8Bx+A+w5/FSCr6+/4lcBhws6qe6P09KCIluM/tTFU9BqgWkaqoPOUDRKx3mYhUiciTuEAXPuHO\nAC709v9o4FQROSjGPt0CXOflzQecIyLTY63rpVsK3Oy9N3qfbo5OKyLrN+FOjOHeHpcAq7zj40+4\n7zR8HPt6yNdcb72jgROB74jIWOBi3EXkTcBrwJ14x7CIjI7x2fSUzn/iTpg5wFpvn/Y5jkWkEncy\nPwYXrG6OsU8KzI04jn/rvf5W5PGjqu+Gv0vgV8BDqvqM995sb933vO/sBOBz3ud1LbAZ+JqqPuht\n/zDg1Ihjown4gvddZwN1wELvs/XhAtxfgaXAAuBr3v/TgB+KSC7wFeBd73u4z8sPwG+AC7z9my8i\nh4jIYcBxqjof+Iz3+cUlLQIAPQ8p0V8Psucqz0/su5Tj9RPg17irgcFoBcpExIe7G7qjn+tHfzYL\ngMki8nfcCfC5eBJR1UeBL3tPp+Gu5M5X1WXeshwvr/E4FVguIo8AjwF/9X68/wN8nb6/x+h9+kyM\nfIRPaMW4Gwm7Y6QT/X134q7szhSRF0XktyJSjPvMlgO3iMhLwDZVrYlK62CgUESeFpFnvav5IuAG\n4HcRed0EnBa+Ao/Ib/Q+HaaqL3mPnwROxp1YTo9e1zs2bsedfBqA70btU6y0EJFPep/LUxHbbQUq\nvcdlwLHsfRzHSutA4AVV7VDVdlzJ4Gjc91ztfcbHAI+y5xgujvHZ9JTOrbir5XO9fWolxnGsqnXA\nwarajSsthI/HyH1ajyvxABzlve8y9pTc9jqOvQuJG4GrvUXhfVqEu3A5A3cMb8MdO2fhjoUbRKTY\nK2HtDywSkVdE5PMAqnqbl8a5QD7uezsMyMWVED7ibSsLeFVVO1U1gDtO5nmvP+Xl6SngZO9CJVdV\n13vLn/a+n48Az3jb3Qxke7+3PqVFAOhlSIn+ptOiqs3eB/kg8J2BpCMin8NdhT3jLRpMYHoVd4Cs\nxv3If96flWN8NtOAelU9BXcyiuuq3UurW0TuwVW3/FFVtwOIyALgKtwPNR5VuB/LJ3FXbA/grg6v\nAZrjyMde+6SqH0bnQ1V34Yr97wFveXmOTifW9/0G8E3vanEd7iRVibsiXYj7wX9dRPaPSq4F+Imq\nnubt0x+ATar6RtQ2u1S1XkR8IvJT4G1VXRPje4o8ZpqBMm/duuh1vTw+7gXBELArYp/+m71/x024\nC4q5uOqF66O29TBwrIis9NZ9Oeo49kWnhQuOx3knvErcybkIGI2rzjgOd9JcgXcMq+qG6M+mh3QK\n1Y0J9iegHFcFdC09HMeqGvSqgV7DVZFE79M3gN+KyO9xpdhv4q7AezqOLwf+rKr13vPwPr2AuwjK\nw33f43FX6PsBfwfe8b6XQtyxdxGuRHZluNSmqn8B7gXGAo/gTvZzvPWvxl24nEnsYXFKgUAvy6KX\n93toHUiTAJBIIjIZd1V8n6o+MMBkPg+c4g1zfQhwr1eUHYiFuCsAiUgrd4BpgTvY/+o9fgw4oj8r\nq+rncO0Ai0SkUETOx10hfsy7AotHLa5+sktd28gk3FXSr3En7dkickt/8hWdD+/HfAzuRDEF+ISI\nHBljvejv+2FVfcd7+WHgUNxn9qaq7lDVFuAl3HcR6X3cSR9V/cBbZ3wPec333lsEXNnDLgUjHpcA\nO3tZ9yLgcu94G4c7OYX36f6otEq9tC4GJnrvuxS4RkROA34K3KKqc3DVLV+OPI5xwXuvtFR1Na66\n9CncBcq/cN9xHe5qeyHwBO4k1OMx3Es6eCfN3+OC6sv0chyr6i9xn/3xXnVj5D6dhvte/4KrDroP\nVz14BvAQ+x7HF+KqicLC+1SLq/KZhQtAVcBRqjoFd5U+Cnfs7AL+T1XbVLXZ+7wPjkjvIlzbw19w\nFwBNqvqi99mGfxvRw+LsxJ3oS3pZBnu+6+jlu4+nvmRUAPBO0s8AC1X1noGmo6rHq+oJXv3hUlwD\n1fYBJlfEnqjegCuiZg00b7iRVc/0Hh+Puyrrk4hc7DXcgisiB4HzcFdMJ2gcDclReTjdS3cCruF1\nrvd5fQZ4T1WviTcxEflsjHwUAa0R1Qk7ibrq6eH7fioiUJyMKz28jVfHLCLZuGqJlVHZ+DxenbO3\nT6XEqP7zqmseBZaq6lciqnOivSMix3uPzwBe6mldVd0/or56B+7kE7lP+6Slqt9S1aO9de7BtXs8\nzd7H21m4kuzu49j7fKLzNRoo9eqdvwLMBl7Hfc8f89IsAjbSyzHcQzr/FJHZuNLMf7DnSnaf41hE\nZonIYm9ZF65BNxi1T8cDk1T1YVwJYxvuhN4BzIk8jr3G5zxVrY7IZnifXsGVYDfiqpH8uMB2JPAh\nrprmLVyAeEVE/F674LHAEhG5VkQu9tIMevl9B9gmIsd6n20IeBH4qIjkefk5EPebfdXLx+7vwWtb\n6BCRGd6xciruYuVV4DSv5DgF8EeUaHqVNr2APIO9bfk63EniehEJ1w2foaptg0x3MH4C3C0iL+N+\nONeqarx17ZHCn024CPwV3EnxwjjXfwi4R0Re9PLxdeBu3A9gsYgAvKiqN/aVkLpeDMeJyBu4H86V\nESdCH/F/jyGvjvW2qHy8oKrfFZFTRORfuHrul1X1H1Hrx/q+vw7cKiKduJPDl7xqomtxdaoAf1LV\n96LSuhP3PYXrxz+vqpFX3uF9+jiuSiRHXM8icN/pP6Pe9w1cKSsXV431lzjWBXfyz4nap6uB/4tI\n6yF6dp233atwv//IyZhCMfL1kLpeQOJ9n0Fc8GkSkUW4UtkpuGqTjcCz7HsMhwNZbQ/p/ABXN34j\nrnT4sLdPex3HqtooIktF5HUvzSdU9SUR2RixT7nAu1HH8e9wx8hexw/u5B2uTw8L79N/40qWDcCf\ncb/Ts3G9hDpwpbCbvGPnPlxA7ATuUdVVIlKHK1Fdias2Oh1XqvgjrrTU6qV1C66q6WXcb+U6VW0X\nkV/jAs7LuEAX/h2Hqx+zgKfV6+3jve91L42eSp37sKEgjDEmQ2VUFZAxxpg9LAAYY0yGsgBgjDEZ\nygKAMcZkKAsAxhiToSwAGGNMhkq3+wCM6ZOITMPdZbkS1188F9iK67tf3cuqQ8q75+FJ3I1HV+me\nMXnCoz1+ynv6uKrGPcSHMT2xAGBGqmpVPTT8xLvZ6Oe4wbmGq0m4O6YnRi4UkZNxN1uFh6d4SkQ+\nrlHDXxvTXxYATKZ4GTcKI7JnTPV83N2ZX8adXD+lqp/xBoJTYKyq1ojIU7g7Q3fihg6uxI0B8zVV\nXSpuAL1K3N2wC3HDB5+Mu/v0UY0Y09/bfiHujtN5uDtif6qqv8PNUzBaRN5Q1aMiVtkKXKOqXd76\nq3HjzBszKNYGYEY8b4yW83FjtuTgRie9SlUPwY2vfj9uzKBjvVVOwo25c7y4iTlmqepbuFv7F6rq\n4bigETmYYI2qzsaNeHm6l/YCYH8RiZ7H4kbv/QcB/wbc6A2GdjawNerkj6q+Fx5Z0wtOn8YNJ2DM\noFgJwIxUE0QkPOpnHm7kyW/jZnSqV9UlAKr6kIjcgRujaLWIzMMNDX0r7kq+BXhO3LjxR+LGAwpv\no0hEKnDtDP/ylm3BjeH/Cu6K/r+9weoinYg3MY43sumj3rZ6nZ1OROZ4aX5DVdf27+MwZl8WAMxI\ntTWyDSDMGy0xmg83uNYTuBEWD8CNVvkCrormb7jfSmtUu8JkdWP/g5viMDynwnzcqJQfA14XkeO9\nIaTD/Ow97r7f236PA3OJyEdwg7xdrap/7n3XjYmPVQGZTKNApYgcASAinwY2qGoDbtjgK4CV3nC6\nnbhqmWdUtRH4QEQu8tY7BRcgIOJkLiIH44b4fUlV/ws3ouasqDw8h5uIJDxE8jleWjEnFhI3p8Ej\nuKkA7eRvEsYCgBmpYl5Ne9Ux5wO/EJHluKFzz/deU+9tL3j/n/cW7/KeXwR8QUTexU1v+emIbYWH\nPH4XNyzvChFZghtu+MmobHwPqBCRZbhgcZOqLu0l39/EdWW9VUTe8f6+1PdHYEzvbDhoY4zJUFYC\nMMaYDGUBwBhjMpQFAGOMyVAWAIwxJkNZADDGmAxlAcAYYzKUBQBjjMlQFgCMMSZD/X9w9Q088wrX\n0AAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x108cf5e48>"
+       ]
+      }
+     ],
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x1 = np.array([1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 100000])\n",
+      "ys = np.array([1, 1, 2, 5, 7, 15, 29, 63, 128, 241, 500, 1016, 2057, 4086, 8259, 16414, 32728, 49956])\n",
+      "y1 = x1 - ys\n",
+      "\n",
+      "\n",
+      "plt.scatter(x1, y1)\n",
+      "plt.xticks(range(19), x1)\n",
+      "plt.xscale('log')\n",
+      "plt.xlabel(\"Powers of 2\")\n",
+      "plt.ylabel(\"Difference\")\n",
+      "plt.title(\"Heads vs Tails\")\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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vOgcyhPqbzhBqvxzYF7g0Ik7JvdABDKH+ZwNfBT5FdgF2UxjC7+8pkuYBdzZZ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\nn2EJ/VszMwAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x108b82a90>"
+       ]
+      }
+     ],
+     "prompt_number": 20
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x1 = np.array([1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 100000])\n",
+      "ys = np.array([1, 1, 2, 5, 7, 15, 29, 63, 128, 241, 500, 1016, 2057, 4086, 8259, 16414, 32728, 49956])\n",
+      "y1 = x1 / ys\n",
+      "\n",
+      "plt.scatter(x1, y1)\n",
+      "plt.xticks(range(19), x1)\n",
+      "plt.xscale('log')\n",
+      "plt.xlabel(\"Powers of 2\")\n",
+      "plt.ylabel(\"Ratio\")\n",
+      "plt.title(\"Heads vs Tails\")\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+       "text": [
+        "<matplotlib.figure.Figure at 0x108ddadd8>"
+       ]
+      }
+     ],
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\"\"\"20 trials of 100000 flips\"\"\"\n",
+      "sides = [0, 1]\n",
+      "\n",
+      "def coin_flip():\n",
+      "    return random.choice(sides)\n",
+      "\n",
+      "coin_flip()\n",
+      "\n",
+      "def flip_counts(n):\n",
+      "    tails = []\n",
+      "    bigger_list = []\n",
+      "    xs = [pow(2, exp) for exp in range(n) if pow(2, exp) < n]\n",
+      "    xs.append(n)\n",
+      "    counter = 0\n",
+      "    while counter <= n:\n",
+      "        coin_flip()\n",
+      "        if coin_flip() == 1:  \n",
+      "            tails.append(1)\n",
+      "        if counter in xs:\n",
+      "            temp_list = tails[:]\n",
+      "            bigger_list.append(temp_list)\n",
+      "        counter += 1\n",
+      "\n",
+      "    ys = [len(i) for i in bigger_list]\n",
+      "    \n",
+      "    return xs, ys\n",
+      "\n",
+      "for i in range(20):\n",
+      "    flip_counts(100000)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "ename": "KeyboardInterrupt",
+       "evalue": "",
+       "output_type": "pyerr",
+       "traceback": [
+        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
+        "\u001b[0;32m<ipython-input-5-ad19ef8da635>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     27\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     28\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m20\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 29\u001b[0;31m     \u001b[0mflip_counts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m100000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+        "\u001b[0;32m<ipython-input-5-ad19ef8da635>\u001b[0m in \u001b[0;36mflip_counts\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m     10\u001b[0m     \u001b[0mtails\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m     \u001b[0mbigger_list\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m     \u001b[0mxs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mpow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mexp\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mpow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     13\u001b[0m     \u001b[0mxs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m     \u001b[0mcounter\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+        "\u001b[0;32m<ipython-input-5-ad19ef8da635>\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     10\u001b[0m     \u001b[0mtails\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m     \u001b[0mbigger_list\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m     \u001b[0mxs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mpow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mexp\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mpow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     13\u001b[0m     \u001b[0mxs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m     \u001b[0mcounter\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+        "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
+       ]
+      }
+     ],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "xs, ys = flip_counts(10000)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "ys = np.array(ys)\n",
+      "x1 = np.array(xs)\n",
+      "y1 = x1 - ys\n",
+      "\n",
+      "plt.scatter(x1, y1)\n",
+      "plt.xticks(range(19), x1)\n",
+      "plt.xscale('log')\n",
+      "plt.xlabel(\"Powers of 2\")\n",
+      "plt.ylabel(\"Difference\")\n",
+      "plt.title(\"Heads vs Tails\")\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+       "text": [
+        "<matplotlib.figure.Figure at 0x112bd5128>"
+       ]
+      }
+     ],
+     "prompt_number": 9
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}
\ No newline at end of file