06. Теория вероятности.ipynb

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      "P(обе | старшая): 0.5007089325501317\n",
      "P(обе | любая из двух):  0.3311897106109325\n"
     ]
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   "source": [
    "# -*- coding: utf-8 -*-\n",
    "\n",
    "# probability.py\n",
    "\n",
    "from collections import Counter\n",
    "import math, random\n",
    "\n",
    "def random_kid():  # произвольно выбрать мальчика или девочку\n",
    "    return random.choice([\"boy\", \"girl\"])\n",
    "\n",
    "def uniform_pdf(x):\n",
    "    return 1 if x >= 0 and x < 1 else 0\n",
    "\n",
    "def uniform_cdf(x):\n",
    "    \"\"\"возвращает вероятность того, что равномерно распределенная\n",
    "    случайная величина <= x\"\"\"\n",
    "    if x < 0:   return 0    # величина никогда не бывает меньше 0\n",
    "    elif x < 1: return x    # например, P(X <= 0.4) = 0.4\n",
    "    else:       return 1    # величина всегда меньше 1\n",
    "\n",
    "def normal_pdf(x, mu=0, sigma=1):\n",
    "    sqrt_two_pi = math.sqrt(2 * math.pi)\n",
    "    return (math.exp(-(x-mu) ** 2 / 2 / sigma ** 2) / (sqrt_two_pi * sigma))\n",
    "\n",
    "def plot_normal_pdfs(plt):\n",
    "    xs = [x / 10.0 for x in range(-50, 50)]\n",
    "    plt.plot(xs,[normal_pdf(x,sigma=1) for x in xs],'-',label='mu=0,sigma=1')\n",
    "    plt.plot(xs,[normal_pdf(x,sigma=2) for x in xs],'--',label='mu=0,sigma=2')\n",
    "    plt.plot(xs,[normal_pdf(x,sigma=0.5) for x in xs],':',label='mu=0,sigma=0.5')\n",
    "    plt.plot(xs,[normal_pdf(x,mu=-1)   for x in xs],'-.',label='mu=-1,sigma=1')\n",
    "    plt.legend()\n",
    "    plt.show()\n",
    "\n",
    "def normal_cdf(x, mu=0,sigma=1):\n",
    "    return (1 + math.erf((x - mu) / math.sqrt(2) / sigma)) / 2\n",
    "\n",
    "def plot_normal_cdfs(plt):\n",
    "    xs = [x / 10.0 for x in range(-50, 50)]\n",
    "    plt.plot(xs,[normal_cdf(x,sigma=1) for x in xs],'-',label='mu=0,sigma=1')\n",
    "    plt.plot(xs,[normal_cdf(x,sigma=2) for x in xs],'--',label='mu=0,sigma=2')\n",
    "    plt.plot(xs,[normal_cdf(x,sigma=0.5) for x in xs],':',label='mu=0,sigma=0.5')\n",
    "    plt.plot(xs,[normal_cdf(x,mu=-1) for x in xs],'-.',label='mu=-1,sigma=1')\n",
    "    plt.legend(loc=4) # снизу справа\n",
    "    plt.show()\n",
    "\n",
    "def inverse_normal_cdf(p, mu=0, sigma=1, tolerance=0.00001):\n",
    "    \"\"\"найти приближенную инверсию, используя двоичный поиск\"\"\"\n",
    "\n",
    "    # если не стандартизировано, то стандартизировать и нормализовать\n",
    "    if mu != 0 or sigma != 1:\n",
    "        return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)\n",
    "\n",
    "    low_z, low_p = -10.0, 0            # normal_cdf(-10) = (очень близко к) 0\n",
    "    hi_z,  hi_p  =  10.0, 1            # normal_cdf(10) = (очень близко к) 1\n",
    "    while hi_z - low_z > tolerance:\n",
    "        mid_z = (low_z + hi_z) / 2     # взять середину\n",
    "        mid_p = normal_cdf(mid_z)      # и значение ИФР в этом месте\n",
    "        if mid_p < p:\n",
    "            # значение середины все еще слишком низкое, искать выше его\n",
    "            low_z, low_p = mid_z, mid_p\n",
    "        elif mid_p > p:\n",
    "            # значение середины все еще слишком высокое, искать ниже\n",
    "            hi_z, hi_p = mid_z, mid_p\n",
    "        else:\n",
    "            break\n",
    "\n",
    "    return mid_z\n",
    "\n",
    "def bernoulli_trial(p):\n",
    "    return 1 if random.random() < p else 0\n",
    "\n",
    "def binomial(p, n):\n",
    "    return sum(bernoulli_trial(p) for _ in range(n))\n",
    "\n",
    "def make_hist(p, n, num_points):\n",
    "\n",
    "    data = [binomial(p, n) for _ in range(num_points)]\n",
    "\n",
    "    # столбчатая диаграмма, показывающая фактические биномиальные выборки\n",
    "    histogram = Counter(data)\n",
    "    plt.bar([x - 0.4 for x in histogram.keys()],\n",
    "            [v / num_points for v in histogram.values()],\n",
    "            0.8,\n",
    "            color='0.75')\n",
    "\n",
    "    mu = p * n\n",
    "    sigma = math.sqrt(n * p * (1 - p))\n",
    "\n",
    "    # линейный график, показывающий нормальное приближение\n",
    "    xs = range(min(data), max(data) + 1)\n",
    "    ys = [normal_cdf(i + 0.5, mu, sigma) - normal_cdf(i - 0.5, mu, sigma)\n",
    "          for i in xs]\n",
    "    plt.plot(xs,ys)\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "if __name__ == \"__main__\":\n",
    "\n",
    "    #\n",
    "    # УСЛОВНАЯ ВЕРОЯТНОСТЬ\n",
    "    #\n",
    "\n",
    "    both_girls = 0\n",
    "    older_girl = 0\n",
    "    either_girl = 0\n",
    "\n",
    "    random.seed(0)\n",
    "    for _ in range(10000):\n",
    "        younger = random_kid()\n",
    "        older = random_kid()\n",
    "        if older == \"girl\":                        # старшая?\n",
    "            older_girl += 1\n",
    "        if older == \"girl\" and younger == \"girl\":  # обе?\n",
    "            both_girls += 1\n",
    "        if older == \"girl\" or younger == \"girl\":   # любая из двух?\n",
    "            either_girl += 1\n",
    "\n",
    "    # обе либо старше\n",
    "    print(\"P(обе | старшая):\",        both_girls / older_girl)   # 0.514 ~ 1/2\n",
    "    # обе либо ни одной\n",
    "    print(\"P(обе | любая из двух): \", both_girls / either_girl)  # 0.342 ~ 1/3\n"
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