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    "### Data Analysis\n",
    "#### Probability: Independent events \n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "For set operaions in Python, see our repository for Python\n",
    "<br> https://github.com/ostad-ai/Python-Everything"
   ]
  },
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    "For indepedent events $A_1$, $A_2$,...,$A_m$ we have:<Br>\n",
    "$P(A_1,A_2,...,A_m)=P(A_1)P(A_2)...P(A_m)$<br>\n",
    "**Special case:** for two independent events $A$ and $B$:<br>\n",
    "$P(A,B)=P(A)P(B)$"
   ]
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     "text": [
      "Experiment: rolling a die\n",
      "Probability of A={1, 3, 5} is 0.5\n",
      "Probability of B={2, 3, 5} is 0.5\n",
      "Probability of A and B: is 0.3333333333333333\n",
      "Two events A={1, 3, 5} and B={2, 3, 5} are not independent.\n"
     ]
    }
   ],
   "source": [
    "#probability of event A with sample space SS\n",
    "def prob(A,SS):\n",
    "    return len(A)/len(SS)\n",
    "\n",
    "# event of intersection A and B \n",
    "def intersection(A,B):\n",
    "    return A&B\n",
    "\n",
    "# Experiment: rolling a die (dice)\n",
    "SS=set(range(1,7))\n",
    "A={1,3,5} # odd numbers in first roll\n",
    "B={2,3,5} #prime numbers in first roll\n",
    "pA=prob(A,SS)\n",
    "pB=prob(B,SS)\n",
    "p_AB=prob(intersection(A,B),SS)\n",
    "p_A,p_B=prob(A,SS),prob(B,SS)\n",
    "print('Experiment: rolling a die')\n",
    "print(f'Probability of A={A} is {p_A}')\n",
    "print(f'Probability of B={B} is {p_B}')\n",
    "print(f'Probability of A and B: is {p_AB}')\n",
    "if p_AB==p_A*p_B:\n",
    "    print(f'Two events A={A} and B={B} are independent.')\n",
    "else:\n",
    "    print(f'Two events A={A} and B={B} are not independent.')"
   ]
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   "source": [
    "Let's examine the probability of systems to work if they are composed of several independent components:<br>\n",
    "- **Series system:** Components are connected in a series form\n",
    "- **Parallel system:** Components are connected in a parallel from"
   ]
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     "text": [
      "We have 3 components with probs. [0.6, 0.3, 0.5] to work correctly.\n",
      "Probability of series network to work: 0.09\n",
      "Probability of parallel network to work: 0.86\n"
     ]
    }
   ],
   "source": [
    "def prob_sys_series(ps):\n",
    "    result=1\n",
    "    for p in ps:\n",
    "        result*=p\n",
    "    return result\n",
    "\n",
    "def prob_sys_parallel(ps):\n",
    "    result=1\n",
    "    for p in ps:\n",
    "        result*=(1-p)\n",
    "    return 1-result\n",
    "\n",
    "# check with what probabilty a system works \n",
    "# if it has two components with the following probabilities to work\n",
    "ps=[.6,.3,.5] #prob. of components\n",
    "print(f'We have {len(ps)} components with probs. {ps} to work correctly.')\n",
    "print(f'Probability of series network to work: {prob_sys_series(ps)}')\n",
    "print(f'Probability of parallel network to work: {prob_sys_parallel(ps)}')"
   ]
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