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    "### Data Analysis\n",
    "#### Conditional probability,  sampling with replacement and without replacement  \n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
   ]
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  {
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    "For set operaions in Python, see our repository for Python\n",
    "<br> https://github.com/ostad-ai/Python-Everything"
   ]
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    "Using **multiplication rule:**<Br>\n",
    "$P(A\\cap B)=P(A)P(B|A)$"
   ]
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     "text": [
      "prob. of being defective:  0.3333333333333333\n",
      "prob. of being non-defective 0.6666666666666666\n",
      "A and B both defective (with replacement) 0.1111111111111111\n",
      "A and B both defective (without replacement) 0.08333333333333333\n"
     ]
    }
   ],
   "source": [
    "#assuming outcomes are equally likely\n",
    "#box contains non-defective products with positive IDs\n",
    "# defective products with negative IDs\n",
    "box={1,2,3,4,5,6,-1,-2,-3}\n",
    "def prob_defective(sample_space,event='defective'):\n",
    "    Ndefective=0\n",
    "    for el in sample_space:\n",
    "        if el<0:\n",
    "            Ndefective+=1  \n",
    "    if event=='defective':        \n",
    "        return Ndefective/len(sample_space)\n",
    "    else:\n",
    "        return (len(sample_space)-Ndefective)/len(sample_space)\n",
    "print('prob. of being defective: ',prob_defective(box,'defective'))\n",
    "print('prob. of being non-defective', prob_defective(box,'nondefective'))\n",
    "print('A and B both defective (with replacement)',\n",
    "prob_defective(box,'defective')*prob_defective(box,'defective'))\n",
    "#assuming first sampling is a defective object\n",
    "# we modify sample_space\n",
    "box2=box.copy()\n",
    "for el in box2:\n",
    "    if el<0:\n",
    "        box2.remove(el)\n",
    "        break\n",
    "print('A and B both defective (without replacement)',\n",
    "prob_defective(box,'defective')*prob_defective(box2,'defective'))"
   ]
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