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karkir0003/PiApproxMonteCarlo (1).ipynb

Last active December 30, 2020 17:44
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PiApproxMonteCarloRevisedForBokeh
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PiApproxMonteCarlo (1).ipynb
{
"nbformat": 4,
"nbformat_minor": 0,
"metadata": {
"colab": {
"name": "PiApproxMonteCarlo.ipynb",
"provenance": [],
"collapsed_sections": []
},
"kernelspec": {
"name": "python3",
"display_name": "Python 3"
},
"widgets": {
"application/vnd.jupyter.widget-state+json": {
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"box_style": "",
"layout": "IPY_MODEL_088133e77e824abfa07e3aa46a80a595",
"_model_module": "@jupyter-widgets/controls",
"children": [
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}
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"_model_module_version": "1.2.0",
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"_model_name": "LayoutModel",
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"height": null,
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},
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"step": 1,
"_model_name": "IntSliderModel",
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"_view_module_version": "1.5.0",
"min": 0,
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"_dom_classes": [],
"_model_name": "OutputModel",
"outputs": [
{
"output_type": "display_data",
"metadata": {
"tags": [],
"needs_background": "light"
},
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zJqGJCMbdMofxdzy1bjucORP23DNpfnrhhZbLW2Xce28y6vHo0clIzzlR9sS6SkjqKulBSY9KekLSGU2UOUrSYkmz0tt32nrcs26fy/KVq9dat3zlas66fW5bd21m7czJTYkigmUrVjJx2oI1Cc64W+YwcdoClq1Y2bovmrfegpNPht12Sy7vtvyJSK5S+/jHkzGFMlb2xLq6vAt8ISJ2AnYGhknas4ly10XEzunt4rYedNHS5a1ab2b54RGKSySJscOHADBx2gImTlsAwKihAxk7fEjpszBPmgQnnOCammqxeHEyptBll8H55ydXr1VYYWINMHb4kDWJ9aihA4mImp4FPJ3i5a30buf01u7VVnUSq5v40VJXw+farFa45qYVChOcBiUnNs8/DyNHwle+4sSmGt11F+y4I5x+Orz7bkUP3fC6GzV0IBOnLWDQaVPWJDatSqyrmKQ6SbOAV4E7ImJ6E8UOkvSYpBsk9WtmX2MkzZA0Y/HixUWP2VRi09x6M8sPJzet0NAUUKiwqaDIg+Dss2HIkA/mPrLq9O67cMYZSZJz330VPXSbEusaEBGrI2JnoC+wu6TtGxW5GRgYETsCdwCXN7OvCRFRHxH1vXv3LnrMPr26tWq9meWHk5sSFfaxGTV0IM+eecCaX9JFE5ylS+HAA+HEE+HttysftLWPp55Kxsb53e8qNv3FOiXWNSgilgJ3A8MarV8SEQ1VahcDn2zrsT7/8aYTn2LrzSw/nNyUSBI9u3ZeqymgoamgZ9fOH/4F/cgjsOuurq2pVatWwY9/nCSvS5e266HWKbGuIZJ6S+qVLncD9gX+3ajMlgV3RwBPtvW4d/+76SarYuvNLD/cobgVTtp327U6bzYkOB9KbC64IKmtqXDfDMvA5MlJEvvXv8In21xZ0KRiiTXQdGJde7YELpdUR/KD7PqIuEXSOGBGREwGTpA0AlgFvAYc1daDvljkqqhi680sP5zctFLjL5K17r/9djJmzdVXVzgqy9Szz8LQofDHPyajS7eDkhPrGhQRjwG7NLF+bMHyacBp5TzuVr26NXnZ91buc2OWe26WKpcnn0xm7XZi0zG9+24ywvQRR7Rb/6pmE2sru1P2G0znurXPcec6ccp+gzOKyMxK5eSmHG6+ORmQb86clstabbv6athjD0+jUSsad2eq7e5NZjXDyU1bXXxxMnaNr4ayBk88Af/v/8Hjj2cdibXBWbfPZeX7a2czK98PT79gVgWc3LTFL3+ZzD+0enXLZa1jWbQIPvvZZJ4qq0ruUGxWvZzcrIsIOP54GDu25bLWcS1dCl/8YjLlhlWdYh2H3aHYLP+c3LTW++/Dt78N552XdSRWDd59F772Nbj22qwjsVY6Zb/BdOtct9a6bp3r3KHYrAo4uWmNVavgG99IJlE0K5VfN1XpwF36cNAn+6yZKLNO4qBP9uHAXfpkHJmZtcTJTalWrYKvf92/wG3dNNT4XXhh1pFYiSbNXMSNDy9aM1Hm6ghufHgRk2YuyjgyM2uJk5tSjR4N//u/WUdh1SwCvvtduOmmrCOxEpx1+1yWr1z7YoHlK1f7aimzKuDkphS/+IWbFKw83n8/aaL617+yjsRa4KulzKqXk5uWXHIJjBuXdRRWS1asgBEjktnFLbd8tZRZ9SppbilJw4CzgTrg4oj4TaPt/YHLgV5pmVMjYkqZY628225rt7mCrINbsgT23x/uvx823zzraKwJp+w3mNNuepxTbj2fIa/OB2A9iY/27g63rZ9xdGY1YOedkzn52kGLyU06E++5wL7AQuAhSZMjonCugZ+RzNR7vqQhwBRgYDvEWzkzZyaX8K5alXUkVqvmz4fhw2HqVNhgg6yjsUYarop6767kY7JLpzr6b9KNzTZ0YmOWd6XU3OwOzIuI+QCSrgVGAoXJTQA90+WNgBfLGWTFPfccfOlL8NZbWUdite6hh+CQQ5KB/urqWi5vFXXgLn1g2o1Zh2FmrVRKn5s+wAsF9xem6wqdDhwhaSFJrc33m9qRpDGSZkiasXjx4nUItwLeeCNpLnjppawjsY7illvg+02+ZczMbB2Uq0PxYcBlEdEXOAC4UtKH9h0REyKiPiLqe/fuXaZDl9lxx8GTT2YdhXU0558P112XdRRmZjWhlORmEdCv4H7fdF2ho4HrASLifqArsFk5AqyoK6+Ea67JOgrrqI49Fp5/PusocklSV0kPSnpU0hOSzmiizPqSrpM0T9J0SQMrH6mZ5UEpyc1DwDaSBknqAhwKTG5U5nlgbwBJnyBJbnLa7lTE/PnJZJhmWVm6NBkDx7PMN+Vd4AsRsROwMzBM0p6NyhwNvB4RWwPjgd9WOEYzy4kWk5uIWAV8D7gdeJLkqqgnJI2TNCIt9kNgtKRHgWuAoyLSMcurwapVcPjh8OabWUdiHd1998Gvf511FLkTiYYe/p3TW+PPmJEkQ1IA3ADsLaUTQ5lZh1LSODfpmDVTGq0bW7A8Bxha3tAq6IwzYPr0rKMwS4wbB/vuC3s2rpjo2NJhKR4GtgbOjYjGb9o1Fz9ExCpJbwCbAv9ptJ8xwBiA/v37t3fYZpYBj1D8z3/6l7LlS8Ms4q5JXEtErI6InUn6/e0uaft13E/+L2wwszbp2MnN0qVwxBHJfD9meeI+YEVFxFLgbmBYo01rLn6Q1IlkzK0llY3OzPKgYyc3p5ziq1Msv668Em6+OesockFSb0m90uVuJCOm/7tRscnAkenywcBdVdX3z8zKpuMmNzNnwqWXZh2FWfN++ENYuTLrKPJgS+BuSY+RXMF5R0Tc0ujChkuATSXNA04GTs0oVjPLWEkdimvSiSe6Ocry7+mn4c9/hpNPzjqSTEXEY8AuTawvvLBhBfC1SsZlZvnUMWtubrgB7r036yjMSjNuHOR1uhIzsxzqeMnNihVJXxuzavHGG/Dzn2cdhZlZ1eh4yc0f/gALFmQdhVnrXHwxPPZY1lGYmVWFjpXcvPQSnHlm1lGYtd7q1XDSSVlHYWZWFTpWcvOTn8Bbb7VcziyP7roLJk3KOgozs9zrOMnN00/DFVdkHYVZ27jvjZlZizpOcnP22b7026rf7Nlwxx1ZR2FmlmsdI7l5/XW47LKsozArjz/8IesIzMxyrWMM4nfhhfD221lHYVYet98Oc+bAkCFZR1LzJs1cxFm3z+XFpcvZqlc3TtlvMAfu0ifrsMysBbVfc7NyJZxzTtZRmJVPBIwfn3UUNW/SzEWcdtPjLFq6nAAWLV3OaTc9zqSZi7IOzcxaUPvJzXXXwSJ/GFmNueoqj1rczs66fS7LV65ea93ylas56/a5GUVkZqWq/eTGv3BbpfEUyp5SOadWrIDzz886ipr24tLlrVpvZvlR28nNPffAI49kHUXVGD/0cMbtPXpNQhPAuL1HM37o4VmGZcWcdx68+27WUdSsrXp1a9V6M8uPmk1uIsK/bFshgGVduzOxfuSaBGfc3qOZWD+SZV27uwYnj155BW68MesoatYp+w2mW+e6tdZ161zHKfsNzigiMytVTVwtNf6Op9h479FEBJKICMZNeoyer3bFA9aXRsDYOy8CYGL9SCbWjwRg1Iy/MfbOi1CGsVkzbrgBDnfNWntouCrKV0uZVZ+qT24igmUrVtKzfiTjbpnD2OFDGHfLHCZOX8io9boQ4C/mEjUkOA2JDel9n78cu+22ZJiD7t2zjqQmHbhLHyczZlWo6pulJDF2+BCWzfgbE6ctYNBpU5g4bQGj3vy3v5hbqaEpqlBhHxzLoeXL4e9/zzqKdiWpn6S7Jc2R9ISkHzRRZi9Jb0iald7GZhGrmeVD1Sc3kCQ4r6dNKg3GXjXOiU0rFPaxGTXjbzz72+GMmvG3tfrgWE7Vfr+bVcAPI2IIsCdwvKSmRjD8Z0TsnN7GVTZEM8uTmkhuIoKNG9c47HaIv5BbQUDPFW+v1cdm7J0XMWrG3+i54m0nink2ZUpSg1OjIuKliHgkXX4TeBJwW5GZFVUTfW7G3TKHnvUjGTV0YNLn5oSz1/QbcdNU6U6a9pe1+ig1JDg+fzn31lvJlAwHHph1JO1O0kBgF2B6E5s/JelR4EXgvyLiiSL7GAOMAejfv3/7BGpmmar65EYSPbt2ZtmMvzH2zAvRqlWMvXoc7HqwaxzWQePzldfz17ijeIfvOH7DDblObiQdERFXSTq5qe0R0eJsoJI2BG4EToyIZY02PwIMiIi3JB0ATAK2KXKsCcAEgPr6elfwmtWgmmiWOmnfbXn9zouQBFOnotdfZ+ydF3HStL9kHZq1g2oZbLCioz3ffDO89157HqGtGi7n6lHk1ixJnUkSm6sj4qbG2yNiWUS8lS5PATpL2qxMsZtZlan6mpsPuftuoIP/iq9hhYMNQtJsVtgROi81OOOHHs6yrt3XNOs1JGA9V7zdPkn3smUwcybssUf5910GEXFh+veM1j5WkoBLgCeL1fBI2gJ4JSJC0u4kP9yWtCFkM6tiJdXcSBomaa6keZJOLVLm6wWXamZXZTK9qaZ4qxWFHZ0n1o9k0I9vWZPY5KV/UGajPVfBa1/SRyXdLGmxpFcl/U3SR1t42FDgm8AXCi71PkDSsZKOTcscDMxO+9z8CTg0ItzkZNZBtVhzI6kOOBfYF1gIPCRpckTMKSizDXAaMDQiXpe0eXsF3Kz334eHHsrk0FY5eR9sMLPRnqsguQH+QvJ58pX0/qHANUDRKqeIuI8WKuQi4hzgnDLFaGZVrpSam92BeRExPyLeA64FRjYqMxo4NyJeB4iIV8sbZomefBLefDOTQ1vlVMNgg4UJToN2T8CqI7nZICKujIhV6e0qoGvWQZlZbSkluekDvFBwfyEfHmNiW2BbSdMkPSBpWFM7kjRG0gxJMxYvXrxuETfnwQfLv0/LlWoZbDCTBOyZZ2BJ7ruZ3CrpVEkDJQ2Q9CNgiqRNJG2SdXBmVhvK1aG4E8lll3sBfYF7Je0QEUsLC7X7JZjV8cvV2qDYYIOk6/PQNNU4ASvs9AztXIPz4IOw//7ttfdy+Hr695hG6w8lOXUt9b8xM2tRKcnNIqBfwf2+6bpCC4HpEbESeFbSUyTJTmU7wDi56RDyPthgpgnY9Om5Tm4iYlDWMZhZ7SsluXkI2EbSIJKk5lCg8YAik4DDgInp2BLbAvPLGWhJHn+84oe0bOR9sMHMEjAn+GZmLfe5iYhVwPeA20nmdLk+Ip6QNE7SiLTY7cASSXOAu4FTIqLyjf+rV1f8kGbFZJKA+WpBM7PS+tykI35OabRubMFyACenNzPLypIlyVxTOSNpaERMk7R+RLybdTxmVttqYvoFMyuwqHGXuFz4U/r3/kyjMLMOofamXzDr6BYuzDqCpqyUNAHoI+lPjTdGxAkZxGRmNcrJjVmtyWfNzXBgH2A/4OGMYzGzGufkxqzW5DC5iYj/ANdKejIiHs06HjOrbe5zY1Zr8tks1WCJpP9NJ818VdKNkvpmHZSZ1ZbaSG58CbjZB3JYc1NgIjAZ2Cq93ZyuMzMrm9pIbl55JesIzPIj38nN5hExsWDizMuA3lkHZWa1pTaSm5deyjoCs/zI9/vhP5KOkFSX3o4Acj/bp5lVl9pIbpYvzzoCs/zI9/vh2ySTZ74MvAQcDIzKNCIzqzm1cbXUqlVZR2CWHzl+P0TEc8CIFgsWkNQPuAL4CMnM4RMi4uxGZQScDRwAvAMcFRGPlCVoM6s6tZHcrFyZdQRm+VF774dVwA8j4hFJPYCHJd0REXMKyuwPbJPe9gDOT/+aWQdUG81SOf6lalZxNfZ+iIiXGmphIuJNkgl8+zQqNhK4IhIPAL0kbVnhUM0sJ2ojuYnIOgKz/Kjh94OkgcAuwPRGm/oALxTcX8iHE6CGfYyRNEPSjMWLF7dHmGaWsdpIbjrVRuuaWVnk+P0g6SOSLpF0a3p/iKSjS3zshsCNwIkRsWxdY4iICRFRHxH1vXv7KnSzWlQbyU3nzllHYJYf+X4/XAbcTjKAH8BTwIktPUhSZ5LE5uqIuKmJIouAfgX3+6brzKwDcnJjVmvy/X7YLCKuB94HiIhVQLNDjKdXQl0CPBkRfyhSbDLwLSX2BN6IiFwP+GNm7Se/9det0aVL1hGY5Ue+3w9vS4mU+NsAABx6SURBVNqU5JJuGhKRFh4zFPgm8LikWem6nwD9ASLiAmAKyWXg80guBffYOWYdWG0kN5tvnnUEZvmx+ebw8stZR1HMySS1LB+TNI1k6oWDm3tARNwHqIUyARxfriDNrLrVRnLTp8mLIsw6pr594bHHso6iSelYNZ8DBpMkLHMjouYG5jGzbNVGcpPvPgZmlZXjZF/Stxqt2lUSEXFFJgGZWU2qjeTGzD6Q4+QG2K1guSuwN/AIyfQKZmZl4eTGrNb07Zt1BEVFxPcL70vqBVybUThmVqNq41JwM/tAvmtuGnsbGJR1EGZWW1xzY1ZrcpzcSLqZ9DJwkh9XQ4Drs4vIzGqRkxuzWpPjZing9wXLq4DnImJhVsGYWW1ycmNWS7p3h403zjqKoiLinqxjMLPa5+TGrJbsumvWETRJ0pt80By11iaSMfh6VjgkM6thJXUoljRM0lxJ8ySd2ky5gySFpPryhdgK22yTyWHNcmOPPbKOoEkR0SMiejZx6+HExszKrcXkRlIdcC6wP0nnv8MkDWmiXA/gB8D0cgdZspx+sJtVTJW8ByRtLql/wy3reMystpRSc7M7MC8i5kfEeyRjUoxsotwvgd8CK8oYX+tUyQe7WbvZffesI2iWpBGSngaeBe4BFgC3ZhqUmdWcUpKbPsALBfcXpuvWkLQr0C8i/t7cjiSNkTRD0ozFixe3OtgW5fyD3axdbbEF9M99JcgvgT2BpyJiEMkIxQ9kG5KZ1Zo2D+InaT3gD8APWyobERMioj4i6nv37t3WQ3/YzjvD+uuXf79m1aA6ai5XRsQSYD1J60XE3UA2ffTMrGaVktwsAvoV3O+brmvQA9gemCppAcmvssmZdCru0iVJcMw6oupIbpZK2hC4F7ha0tkkoxSbmZVNKcnNQ8A2kgZJ6gIcCkxu2BgRb0TEZhExMCIGklQxj4iIGe0ScUuq4wPerPyq47U/EngHOAm4DXgG+HKmEZlZzWkxuYmIVcD3gNuBJ4HrI+IJSeMkjWjvAFvt05/OOgKzyuvSBXbbreVy2TsG2DIiVkXE5RHxp7SZqlmSLpX0qqTZRbbvJekNSbPS29iyR25mVaOkQfwiYgowpdG6Jj88ImKvtofVBvvvD127worsLtoyq7h99oEePbKOohQ9gH9Ieg24DvhrRLxSwuMuA84BrmimzD8jYnjbQzSzald7s4JvuCHst1/WUZhV1sEHZx1BSSLijIjYDjge2BK4R9L/lfC4e4HX2js+M6sNtZfcABx0UNYRmFVOp04wsqmhp3LtVeBlYAmweZn2+SlJj0q6VdJ2xQq1+5AUZpa52kxuRoxI+iCYdQSf/zxssknWUZRE0nclTQXuBDYFRkfEjmXY9SPAgIjYCfgzMKlYwXYfksLMMlebyc1GG8Hee2cdhVllVFdNZT/gxIjYLiJOj4g55dhpRCyLiLfS5SlAZ0mblWPfZlZ9ajO5garpg2DWJnV18JWvZB1FySLitIiYVe79StpCktLl3Uk+21q8CsvMalPtJjcHHpj0RTCrZZ/5DGxeri4r+SXpGuB+YLCkhZKOlnSspGPTIgcDsyU9CvwJODQiIqt4zSxbtfvtv8kmSV+EO+7IOhKz9lNdTVLrLCIOa2H7OSSXipuZ1XDNDcB3vpN1BGbtZ4MN4PDDs47CzCx3aju5OeggGDAg6yjM2seRR1bNVVJmZpVU28lNXR18//tZR2FWfhKcdFLWUZiZ5VJtJzcAo0dXy7D0ZqUbPhy22SbrKMzMcqn2k5uePeHoo7OOwqy8Tj456wjMzHKr9pMbgB/8IGmiMqsFu+4Ke+2VdRRmZrnVMZKbgQOraqAzs2a5r42ZWbM6RnIDrsa32rDVVnDIIVlHYWaWax0nufnUpzzflFW/U0+Fzp2zjsLMLNc6TnIDMH68+95Y9frEJ+C447KOwsws9zpWcrPDDsml4WbV6A9/8HxpZmYl6FjJDcAvfwm9emUdhVnrHHAADBuWdRRmZlWh4yU3m20GP/951lGYla5Tp6TWxszMStLxkhtIpmTYdtusozArzfHHw+DBWUdhZlY1OmZy07kz/M//ZB2FWcs23RR+8YusozAzqyodM7mBZG6eL34x6yjMmnfGGbDxxllHYWZWVTpucgNw3nmeVNPya+hQOPbYrKPIBUmXSnpV0uwi2yXpT5LmSXpM0q6VjtHM8qNjJzcf+xj8+c9ZR2H2YRttBFdf7XGZPnAZ0NzlYvsD26S3McD5FYjJzHKqYyc3AEce6eHsLX/OPx8GDMg6ityIiHuB15opMhK4IhIPAL0kbVmZ6Mwsb5zcAFxwgb9ILD+++U047LCso6g2fYAXCu4vTNd9iKQxkmZImrF48eKKBGdmleXkBpJB/a66yk0Alr2PfQzOPTfrKGpaREyIiPqIqO/du3fW4ZhZOygpuZE0TNLctLPeqU1sP1nSnLQj352Sqq8a5NOfhp/8JOsorCPr1CnpZ+NO7utiEdCv4H7fdJ2ZdUAtJjeS6oBzSTrsDQEOkzSkUbGZQH1E7AjcAPyu3IFWxNixsOeeWUdhHdUvfgF77JF1FNVqMvCt9KqpPYE3IuKlrIMys2yUUnOzOzAvIuZHxHvAtSSd99aIiLsj4p307gMkv5qqT6dO8Je/JAOnmVXSF77gmsNmSLoGuB8YLGmhpKMlHSup4Vr5KcB8YB5wEfDdjEI1sxwoZYrhpjrqNffz8mjg1qY2SBpDcpkm/fv3LzHEChs0CP72N9hnH1ixIutorCPYbju46SZYz13giomIZntYR0QAx1coHDPLubJ+mko6AqgHzmpqe9V05Bs6NOlg7C8ba29bbQW33pqMa2NmZmVRyrd3SR31JO0D/BQYERHvlie8DB10kOefsvbVowdMmQL9+rVc1szMSlZKcvMQsI2kQZK6AIeSdN5bQ9IuwIUkic2r5Q8zIyeemNzMyq1zZ7jxRthpp6wjMTOrOS0mNxGxCvgecDvwJHB9RDwhaZykEWmxs4ANgb9KmiVpcpHdVZ//+Z+kFsesnC66CPbdN+sozMxqUikdiomIKSRXIxSuG1uwvE+Z48qP9dZL+t+8/DJMm5Z1NFYLzjgjmfbDzMzahXvMlqJrV5g8GbbfPutIrNode2wynpKZmbUbJzel2mQTmDoVdt0160isWp1wQjIhppmZtSsnN62x6aZw550exdha78c/hrPPzjoKM7MOwclNa/XqBXfcAZ/7XNaRWLU44wz4zW+yjsLMrMNwcrMuNtwQbrsNvvrVrCOxPFtvPTjvPPexMTOrMCc366prV/jrX5MOomaNrb9+8vo47risIzEz63Cc3LTFeuslHUTHjcs6EsuTXr3gH/9wzZ6ZWUac3JTDz3+eTHzo+YFs551hxgz47GezjsTMrMNyclMuX/kKPPxw8uVmHdPo0XD//fCxj2UdiZlZh+bkppw+9rHky2306KwjsUraYAO4/HKYMCHpi9UGEdHsfTMza5mTm3Lr2jX5krv88uRLz2rb4MEwfTp861tt3tX4O55i3C1z1iQ0EcG4W+Yw/o6n2rxvM7OOxMlNe/nWt5IvvcGDs47E2sshhyT9a8owLUdEsGzFSiZOW7AmwRl3yxwmTlvAshUrO3wNjqRhkuZKmifp1Ca2HyVpcTpx7yxJ38kiTjPLh5ImzrR1tP32yZffd78LV16ZdTRWLt27w+9+l/xfy0QSY4cPAWDitAVMnLYAgFFDBzJ2+BAkle1Y1UZSHXAusC+wEHhI0uSImNOo6HUR8b2KB2hmueOam/a24YZwxRXJpcFbb511NNZWX/4yzJlT1sSmQWGC06CjJzap3YF5ETE/It4DrgVGZhyTmeWYk5tK2XdfePzx5LLxLl2yjsZaq2/f5HL/yZOhf/92OURDU1Shwj44HVgf4IWC+wvTdY0dJOkxSTdI6ldsZ5LGSJohacbixYvLHauZ5YCTm0rq2jUZ8O/RRz03VbWoq4MTT0xqa77ylXY7TGEfm1FDB/LsmQcwaujAtfrgWLNuBgZGxI7AHcDlxQpGxISIqI+I+t69e1csQDOrHCc3Wfj4x2HqVLjsMthss6yjsWLq6+HBB2H8eOjRo10PJYmeXTuv1cdm7PAhjBo6kJ5dO3f0pqlFQGFNTN903RoRsSQi3k3vXgx8skKxmVkOuUNxlo48EoYPh5/+FC69FFauzDoiA9h882Syy+OOS6bYqJCT9t2WiFiTyDQkOB08sQF4CNhG0iCSpOZQ4PDCApK2jIiX0rsjgCcrG6KZ5YlrbrK26aZwwQXw1FPJ4H+dO2cdUcf1kY/A738Pzz4Lxx/fpsRmXQfja5zIOLGBiFgFfA+4nSRpuT4inpA0TtKItNgJkp6Q9ChwAnBUNtGaWR4oq7b8+vr6mDFjRtn2J6k2+iU89xz8+tcwcaJrcipliy3gRz9KZnjv1q3Nuxt/x1MsW7FyTa1LQ3+anl07c9K+25Yh4JaV+/0g6eGIqC/bDnOi3J9DZtY+WvsZ5JqbvBkwAC68EObNg2OO8ZVVQOOv6LJ9ZW+xRdKfZv58OOmksiQ2HozPzCx77nOTV/37J81VP/0pnHlmMp3DO+9kHVXFjR96OMu6dmfsnRchksRm3N6j6bnibU6a9pd122nfvvBf/wVjxpQloSnkwfjMzLLnmpu869cPzjsPXnopSXZ22y3riComgGVduzOxfiTj9h69JrGZWD+SZV27t64Gp0sXOPhguPXWpOnvBz8oe2LTwIPxmZllyzU31aJnz6SZ6phjksEAL700mdJhyZKsI2s3AsbeeREAE+tHMrE+GZR21Iy/ranJadF228HRR8M3v1mxy+6LDcbnBMfMrDJcc1NF1vTX2GEHGD+eWLQIrr8e9tuvopcsV1JhgtOgxcSmZ8+kyWn6dJg9O+lPU+HExoPxmZllxzU3VaLJK3D+MY+evXbipNtug4UL4eabk2aXu+6Ct9/OOuSyaGiKKjRu79EfTnD69oVhw5Lb/vvDBhtUMsw1ig3GB3gwPjOzCnFyUwUKr8CBpP9GYe1ARKC+fZNB5447Dt57D/75T7jttuQ2e3a2T2AdFfaxaWiKarjPenWMXf0Uakhmtt8+63DX8GB8ZmbZKim5kTQMOBuoAy6OiN802r4+cAXJkOdLgEMiYkF5Q+24Wn0FTpcusPfeye2ss5JanYZE51//SjonVwEBPVe8nSQ2d1+CPv5xxm7bCbZ8n54/OB59KT8JTWMejM/MLDstDuInqQ54CtiXZDbeh4DDImJOQZnvAjtGxLGSDgW+EhGHNLdfD+LXehHBoNOmrLn/7JkHrNuX5quvwqxZyQSes2Ylt7lzYfXqMkbbBt27w447ws47w047ETvthHbccU1TU2GtiDXNg/iVxoP4mVWH1n4GlVJzszswLyLmpwe4FhgJFF4OMhI4PV2+AThHkqLWs40KKusVOJtvDl/8YnJrsGJF0nw1a1YyA/aLL8Irr8DLLye3pUvL8CxSdXXQu3cy3cEWWyS3fv2ShGannWDrrdfqIN342TmxMTOz5pSS3PQBXii4vxDYo1iZiFgl6Q1gU+A/5Qiyo2t8BU5hnxso0xgqXbsms2DXF0mM3333g2Sn4e+bb8KqVWvfJOjU6YNb586wySYfJDEf+UiS2NTo1V1mZpa9inYoljQGGAPQv3//su57wIABNf2LfqOhh7Ne1+6c/tuL1lSRbbz3aMbf9zanj1jHkXqtZg0YMCDrEMzMMlNKcrMI6Fdwv2+6rqkyCyV1AjYi6Vi8loiYAEyApK17XQIuZsGCBeXcXS4lfU0mNLov4OrsgjIzM8uZUtoGHgK2kTRIUhfgUGByozKTgSPT5YOBu9zfpvx8BY6ZmVnLWqy5SfvQfA+4neRS8Esj4glJ44AZETEZuAS4UtI84DWSBMjMzMys4krqcxMRU4ApjdaNLVheAXytvKGZmZmZtZ4vWTGz3JM0TNJcSfMkndrE9vUlXZduny5pYDmOO2nmIob+5i4Gnfp3hv7mLibNbNzd0MzyyMmNmeVaOpDoucD+wBDgMElDGhU7Gng9IrYGxgO/betxJ81cxGk3Pc6ipcsJYNHS5Zx20+NOcMyqgJMbM8u7NQOJRsR7QMNAooVGApenyzcAe6uNPe7Pun0uy1euPWr38pWrOev2uW3ZrZlVgJMbM8u7pgYS7VOsTESsAhoGEv0QSWMkzZA0Y/HixUUP+uLS5a1ab2b54eTGzDqUiJgQEfURUd+7d++i5bbq1a1V680sP5zcmFnetWYgUZobSLQ1TtlvMN061621rlvnOk7Zb3BbdmtmFeDkxszyLpOBRA/cpQ9nfnUH+vTqhoA+vbpx5ld34MBdGreImVneVHRuKTOz1spyINEDd+njZMasCjm5MbPc80CiZtYabpYyMzOzmuLkxszMzGqKkxszMzOrKU5uzMzMrKaojVdLrvuBpcXAc2Xc5WbAf8q4v3JybOsmz7FBvuMrd2wDIqL4iHdVqhWfQ3n+XzfmWNuHYy2/1sTZqs+gzJKbcpM0IyLqs46jKY5t3eQ5Nsh3fHmOrRpV0/l0rO3DsZZfe8bpZikzMzOrKU5uzMzMrKbUUnIzIesAmuHY1k2eY4N8x5fn2KpRNZ1Px9o+HGv5tVucNdPnxszMzAxqq+bGzMzMzMmNmZmZ1ZaqSm4kDZM0V9I8Sac2sX19Sdel26dLGpiz+E6WNEfSY5LulDQgL7EVlDtIUkiq2GWEpcQm6evpuXtC0l/yEpuk/pLuljQz/b8eUMHYLpX0qqTZRbZL0p/S2B+TtGulYqslpb532vH4/dLXWMPr/wfp+k0k3SHp6fTvxun6ov93SUem5Z+WdGQ7xlyXviduSe8PSj+T56Wf0V3S9UU/syWdlq6fK2m/doqzl6QbJP1b0pOSPpXX8yrppPT/P1vSNZK65uW8NvVZVM7zKOmTkh5PH/MnSWoxqIioihtQBzwDfBToAjwKDGlU5rvABenyocB1OYvv88AG6fJxlYqvlNjScj2Ae4EHgPq8xAZsA8wENk7vb56j2CYAx6XLQ4AFFXzNfRbYFZhdZPsBwK2AgD2B6ZWKrVZupb532jmGLYFd0+UewFPpa+13wKnp+lOB3zb3fwc2AeanfzdOlzdup5hPBv4C3JLevx44NF2+oOA90+Rndvr8HgXWBwal/4O6dojzcuA76XIXoFcezyvQB3gW6FZwPo/Ky3lt6rOonOcReDAtq/Sx+7cUUzXV3OwOzIuI+RHxHnAtMLJRmZEkL1aAG4C9S8rwKhRfRNwdEe+kdx8A+uYlttQvgd8CKyoUV6mxjQbOjYjXASLi1RzFFkDPdHkj4MUKxUZE3Au81kyRkcAVkXgA6CVpy8pEVzNKfe+0m4h4KSIeSZffBJ4k+bIr/Ly7HDgwXS72f98PuCMiXkvfS3cAw8odr6S+wJeAi9P7Ar5A8pncVKxNfWaPBK6NiHcj4llgHsn/opxxbkTypXwJQES8FxFLyel5BToB3SR1AjYAXiIn57XIZ1FZzmO6rWdEPBBJpnNFwb6Kqqbkpg/wQsH9hem6JstExCrgDWDTikRXWnyFjibJQCuhxdjSqsF+EfH3CsXUoJTzti2wraRpkh6Q1B4fHOsa2+nAEZIWAlOA71cmtJK09jVpH5arc5g2L+wCTAc+EhEvpZteBj6SLheLuVLP5Y/Aj4D30/ubAkvTz+TGxy32mV2JWAcBi4GJaRPaxZK6k8PzGhGLgN8Dz5MkNW8AD5PP89qgXOexT7rceH2zqim5qRmSjgDqgbOyjgVA0nrAH4AfZh1LEZ1Imqb2Ag4DLpLUK9OIPnAYcFlE9CWpbr0yPZ9mZSVpQ+BG4MSIWFa4Lf1Fm/m4HpKGA69GxMNZx1KCTiRNKedHxC7A2yTNJ2vk6LxuTFLjMQjYCuhO+9QOtYsszmM1fQgvAvoV3O+brmuyTFp1txGwpCLRlRYfkvYBfgqMiIh3cxJbD2B7YKqkBSRtm5NVmU7FpZy3hcDkiFiZVqU+RZLs5CG2o0navYmI+4GuJJPB5UFJr0lrVi7OoaTOJInN1RFxU7r6lYZmxvRvQ3NtsZgr8VyGAiPSz5FrSZpNziZpeujUxHGLfWZXItaFwMKImJ7ev4Ek2cnjed0HeDYiFkfESuAmknOdx/PaoFzncRFrd+EoLea2diSq1I0ky55Pkrk2dOzbrlGZ41m7E9X1OYtvF5IOXNvk7dw1Kj+VynUoLuW8DQMuT5c3I6m63DQnsd0KHJUuf4Kkz40q+L8dSPEOxV9i7Y57D1bydVcLt9a+d9opBpH0M/hjo/VnsXaHzd81938n6aj5LElnzY3T5U3aMe69+KBD8V9Zu+Prd9PlJj+zge1Yu+PrfNqnQ/E/gcHp8unpOc3deQX2AJ4g6Wsjkj4s38/TeW38WVTO88iHOxQf0GI87fXCbqc3ywEkv9qfAX6arhtHUgsCya/mv5J0knoQ+GjO4vs/4BVgVnqbnJfYGpWdSoWSmxLPm0iazeYAjze8mXMS2xBgWvqBMQv4YgVju4ak/X0lya/Qo4FjgWMLztu5aeyPV/J/Wku3pl4DFT7+p0mq9B8r+Ow4gKQPxZ3A0+lnS8MXQdH/O/Dt9PNxHjCqnePeiw+Sm4+mn8nz0s/o9dP1RT+zSWq4nwHmUsLVMesY487AjPTcTiL5Us3leQXOAP4NzAauJElQcnFei3wWle08knTjmJ0+5hxK+AHp6RfMzMysplRTnxszMzOzFjm5MTMzs5ri5MbMzMxqipMbMzMzqylObszMzKymOLkxM7OaIGlqWwcflTRCGcz+buXVqeUiZmZmHUNETAYmZx2HtY1rbszMrF1I6i7p75IelTRb0iHp+rGSHkrXTUhnrG6oeRkvaYakJyXtJukmSU9L+lVaZqCkf0u6Oi1zg6QNmjj2FyXdL+kRSX9N5+ZqXOYESXMkPSbp2nTdUZLOSZdnFdyWS/pc+pwulfRgOuFmRWeJt9I4uTEzs/YyDHgxInaKiO2B29L150TEbum6bsDwgse8FxH1JNMJ/I1kKoHtgaMkbZqWGQycFxGfAJYB3y08qKTNgJ8B+0TEriSjEJ/cRHynArtExI4ko3uvJSJ2joidgZ+n+/gXyQi/d0XE7sDngbPS2cQtR5zcmJlZe3kc2FfSbyV9JiLeSNd/XtJ0SY+TTK65XcFjJhc89omIeCmSSYbn88HEii9ExLR0+SqSKSoK7Uk6NYqkWcCRwIAm4nsMuFrSEcCqpp6ApG1I5kn6eiSTVn4RODXd71SSqQ76t3QirLLc58bMzNpFRDwlaVeSubB+JelO4HfAeSRzCr0g6XSSBKHBu+nf9wuWG+43fGc1njeo8X0Bd0TEYS2E+CXgs8CXgZ9K2mGtnSRNWdcDoyPipYJ9HxQRc1vYt2XINTdmZtYuJG0FvBMRV5HUfuzKB4nMf9Lk4eB12HV/SZ9Klw8H7mu0/QFgqKSt0zi6S9q2UWzrAf0i4m7gx8BGQON+OZcCEyPinwXrbge+X9BPaJd1iN/amWtuzMysvexA0iflfZIZo4+LiKWSLiKZ5fll4KF12O9c4HhJlwJzgPMLN0bEYklHAddIWj9d/TOS2d0b1AFXSdqIpDbmT2lsAEgaQJJ4bSvp2+ljvgP8Evgj8FiaID3L2n2GLAc8K7iZmVUNSQOBW9LOyGZNcrOUmZmZ1RTX3JiZmVlNcc2NmZmZ1RQnN2ZmZlZTnNyYmZlZTXFyY2ZmZjXFyY2ZmZnVlP8Prczwj0L8SS8AAAAASUVORK5CYII=\n",
"text/plain": "<Figure size 576x288 with 2 Axes>"
}
],
"_view_module": "@jupyter-widgets/output",
"_model_module_version": "1.0.0",
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"source": [
"#Monte Carlo Method to Approximate Pi\r\n",
"##By Karthik Subramanian and Valentin Stelea\r\n",
"\r\n",
"This notebook takes you through the process of using repeated random sampling to approximate $\\pi$. The user is able to play around with the interactive plots and test out different parameters to gain an understanding of how this method can be used to approximate $\\pi$."
]
},
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"id": "ahD1OVUJS7WL"
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"#function that uses unit square with a circle of radius 0.5 inside\r\n",
"def getPiApproximationMonteCarlo(N_samples):\r\n",
" x = np.random.uniform(0,1,N_samples)\r\n",
" y = np.random.uniform(0,1,N_samples)\r\n",
" # check if in circle\r\n",
" check = ((x-0.5)**2+(y-0.5)**2 <=1/4).astype(int)\r\n",
" freq = np.sum(check)/N_samples\r\n",
" return freq*4"
],
"execution_count": 1,
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"\"\"\"\r\n",
"Interactive widget that allows the user to adjust the sample size to see how the value of the approximated pi changes\r\n",
"\"\"\"\r\n",
"\r\n",
"# nbi:hide_in\r\n",
"import matplotlib.pyplot as plt\r\n",
"import numpy as np\r\n",
"from ipywidgets import interact\r\n",
"import math\r\n",
"\r\n",
"plt.rcParams[\"figure.figsize\"] = (8, 4)\r\n",
"\r\n",
"# generate data\r\n",
"N_samples = 10000\r\n",
"x = np.random.uniform(0, 1, N_samples)\r\n",
"y = np.random.uniform(0, 1, N_samples)\r\n",
"\r\n",
"# check if in circle\r\n",
"check = ((x-0.5)**2+(y-0.5)**2 <=1/4).astype(int)\r\n",
"piVals = np.array([ getPiApproximationMonteCarlo(i) for i in range(N_samples) ])\r\n",
"\r\n",
"# ploting results\r\n",
"def draw(i=10):\r\n",
" fig, [ax1, ax2] = plt.subplots(1,2, num=1, clear=True)\r\n",
" draw_circle = plt.Circle((0.5, 0.5), 0.5, facecolor='red')\r\n",
" draw_square = plt.Rectangle((0, 0), width=1, height=1, fill=False, zorder=1)\r\n",
" ax1.add_artist(draw_circle)\r\n",
" ax1.add_artist(draw_square)\r\n",
" ax1.set_xlim(-0.1, 1.1)\r\n",
" ax1.set_ylim(-0.1, 1.1)\r\n",
" ax1.set_axis_on()\r\n",
" ax1.title.set_text('Unit Square with circle of radius 0.5')\r\n",
" ax1.scatter(x[:i], y[:i], marker=\"x\", zorder=2)\r\n",
" \r\n",
" prob = np.pi/(4*4)\r\n",
" tosses = np.arange(i)\r\n",
" ax2.scatter(tosses, piVals[:i])\r\n",
" ax2.set_xlabel('sample size')\r\n",
" ax2.set_ylabel('value of pi')\r\n",
" ax2.plot([0,N_samples], [math.pi,math.pi], 'r')\r\n",
" plt.tight_layout()\r\n",
" plt.show()\r\n",
"\r\n",
"interact(draw, i=(0,N_samples, 1));"
],
"execution_count": 2,
"outputs": [
{
"output_type": "stream",
"text": [
"/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:7: RuntimeWarning: invalid value encountered in long_scalars\n",
" import sys\n"
],
"name": "stderr"
},
{
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"model_id": "9774d4d3fa694eefa1f47ea14aa2c564",
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"interactive(children=(IntSlider(value=10, description='i', max=10000), Output()), _dom_classes=('widget-intera…"
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{
"cell_type": "markdown",
"metadata": {
"id": "mfC1-d84tXxe"
},
"source": [
"Bokeh Distribution Plot for Pi Approximation. "
]
},
{
"cell_type": "code",
"metadata": {
"id": "TYDTFBjCty_b"
},
"source": [
"import pandas as pd\r\n",
"from bokeh.plotting import figure, show\r\n",
"from bokeh.io import output_notebook\r\n",
"from bokeh.models import ColumnDataSource, HoverTool\r\n",
"def generateDataFrame(num_iterations, sample_size):\r\n",
" pi_samples = [getPiApproximationMonteCarlo(sample_size) for i in range(num_iterations)] #100 samples\r\n",
" pi_samples = sorted(pi_samples, reverse=False)\r\n",
" countDict = {}\r\n",
" for element in pi_samples:\r\n",
" # checking whether it is in the dict or not\r\n",
" if element in countDict:\r\n",
" # incerementing the count by 1\r\n",
" countDict[element] += 1\r\n",
" else:\r\n",
" # setting the count to 1\r\n",
" countDict[element] = 1\r\n",
" pi_frame = pd.DataFrame.from_dict({'pi-value': countDict.keys(), 'count': countDict.values()})\r\n",
" pi_frame[\"adjusted\"] = pi_frame['pi-value'] + 0.001\r\n",
" return pi_frame\r\n",
"\r\n",
"\r\n",
"def bokehPlot(pi_frame):\r\n",
" # Convert dataframe to column data source\r\n",
" src = ColumnDataSource(pi_frame)\r\n",
" # Create the blank plot\r\n",
" p = figure(plot_height = 600, plot_width = 600, \r\n",
" title = 'Histogram of Values of Pi from Simulation',\r\n",
" x_axis_label = 'Values of Pi from Apporximation', \r\n",
" y_axis_label = 'Counts')\r\n",
"\r\n",
" # Add a quad glyph\r\n",
" p.quad(source=src, bottom=0, top='count',\r\n",
" left='pi-value', right='adjusted', \r\n",
" fill_color='red', line_color='black')\r\n",
" # Hover tool referring to our own data field using @ and\r\n",
" # a position on the graph using $\r\n",
" h = HoverTool(tooltips = [('Pi Value', '@{pi-value}'),\r\n",
" ('Count', '@count')])\r\n",
" p.add_tools(h)\r\n",
"\r\n",
" \r\n",
" # Show the plot\r\n",
" show(p)\r\n",
"\r\n",
"\r\n",
"def runMonteCarlo(num_iterations, sample_size):\r\n",
" df = generateDataFrame(num_iterations, sample_size)\r\n",
" bokehPlot(df)"
],
"execution_count": 3,
"outputs": []
},
{
"cell_type": "code",
"metadata": {
"id": "6EsFBhnEjInc"
},
"source": [
"output_notebook()"
],
"execution_count": 4,
"outputs": []
},
{
"cell_type": "code",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 650
},
"id": "lsrprI50u4Hk",
"outputId": "fd93dfb2-b385-416e-88a8-6c6b401bbd16"
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"source": [
"sample_size = int(input(\"How many elements would you like to sample?\"))\r\n",
"\r\n",
"num_iterations = int(input(\"How many times would you like to sample?\"))\r\n",
"\r\n",
"runMonteCarlo(num_iterations, sample_size)\r\n"
],
"execution_count": 5,
"outputs": [
{
"output_type": "stream",
"text": [
"How many elements would you like to sample?1000\n",
"How many times would you like to sample?1000\n"
],
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" var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.1.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.1.1.min.js\"];\n",
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" } else {\n",
" attempts++;\n",
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"})(window);"
],
"application/vnd.bokehjs_exec.v0+json": ""
},
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]
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]
}
Raw
requirements.txt
# Automatically generated by https://github.com/damnever/pigar.
# C:\Users\karki\OneDrive\Desktop\pi approx monte carlo\PiApproxMonteCarlo (2).ipynb: 56
bokeh == 1.3.4
# C:\Users\karki\OneDrive\Desktop\pi approx monte carlo\PiApproxMonteCarlo (2).ipynb: 16
ipywidgets == 7.5.1
# C:\Users\karki\OneDrive\Desktop\pi approx monte carlo\PiApproxMonteCarlo (2).ipynb: 14
matplotlib == 3.1.1
# C:\Users\karki\OneDrive\Desktop\pi approx monte carlo\PiApproxMonteCarlo (2).ipynb: 15
numpy == 1.16.5
# C:\Users\karki\OneDrive\Desktop\pi approx monte carlo\PiApproxMonteCarlo (2).ipynb: 53
pandas == 0.25.1
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