{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "a1b2c3d4-1111-4222-8333-944455556666", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Хоккей \u03bb = 3.2 : 2.8 P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442) = 56.4% P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440) = 43.6% CV = 59.8%\n", "\u0411\u0430\u0441\u043a\u0435\u0442\u0431\u043e\u043b \u03bb = 125.7 : 110.0 P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442) = 84.7% P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440) = 15.3% CV = 9.5%\n" ] } ], "source": [ "import numpy as np\n", "from scipy.stats import poisson\n", "\n", "def match_probs(lam_fav, lam_dog, max_goals=250):\n", " k = np.arange(max_goals)\n", " p_fav = poisson.pmf(k, lam_fav)\n", " p_dog = poisson.pmf(k, lam_dog)\n", " joint = np.outer(p_fav, p_dog) # joint[i, j] = P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442=i, \u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440=j)\n", " win = np.tril(joint, k=-1).sum() # \u0444\u0430\u0432\u043e\u0440\u0438\u0442 \u0437\u0430\u0431\u0440\u043e\u0441\u0438\u043b \u0431\u043e\u043b\u044c\u0448\u0435\n", " tie = np.trace(joint)\n", " lose = np.triu(joint, k=1).sum() # \u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440 \u0437\u0430\u0431\u0440\u043e\u0441\u0438\u043b \u0431\u043e\u043b\u044c\u0448\u0435\n", " return win, tie, lose\n", "\n", "# \u0422\u043e\u0442 \u0436\u0435 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0439 \u0440\u0430\u0437\u0440\u044b\u0432 \u0432 \u043a\u043b\u0430\u0441\u0441\u0435 (8:7), \u0440\u0430\u0437\u043d\u044b\u0439 \u043c\u0430\u0441\u0448\u0442\u0430\u0431 \u0441\u0447\u0451\u0442\u0430\n", "lam_fav_h, lam_dog_h = 3.2, 2.8 # \u0445\u043e\u043a\u043a\u0435\u0439: \u0448\u0430\u0439\u0431\u044b \u0437\u0430 \u043c\u0430\u0442\u0447\n", "lam_dog_b = 110.0\n", "lam_fav_b = lam_dog_b * (lam_fav_h / lam_dog_h) # \u0431\u0430\u0441\u043a\u0435\u0442\u0431\u043e\u043b: \u043e\u0447\u043a\u0438 \u0437\u0430 \u043c\u0430\u0442\u0447\n", "\n", "for sport, lf, ld in [('\u0425\u043e\u043a\u043a\u0435\u0439', lam_fav_h, lam_dog_h),\n", " ('\u0411\u0430\u0441\u043a\u0435\u0442\u0431\u043e\u043b', lam_fav_b, lam_dog_b)]:\n", " win, tie, lose = match_probs(lf, ld)\n", " p_fav = win + tie / 2 # \u043d\u0438\u0447\u044c\u044f \u0441\u043d\u0438\u043c\u0430\u0435\u0442\u0441\u044f \u043e\u0432\u0435\u0440\u0442\u0430\u0439\u043c\u043e\u043c/\u0431\u0443\u043b\u043b\u0438\u0442\u0430\u043c\u0438 - \u043c\u043e\u043d\u0435\u0442\u043a\u043e\u0439\n", " cv = ld ** 0.5 / ld # \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u0438 \u0441\u0447\u0451\u0442\u0430 \u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440\u0430\n", " print(f'{sport:10s} \u03bb = {lf:.1f} : {ld:.1f} '\n", " f'P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442) = {p_fav:.1%} P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440) = {1 - p_fav:.1%} '\n", " f'CV = {cv:.1%}')" ] }, { "cell_type": "code", "execution_count": 2, "id": "b2c3d4e5-2222-4333-8444-955566667777", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Хоккей MC P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, 1 \u043c\u0430\u0442\u0447) = 56.4% P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, \u0441\u0435\u0440\u0438\u044f \u0434\u043e 4 \u043f\u043e\u0431\u0435\u0434) = 63.7% P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440, \u0441\u0435\u0440\u0438\u044f) = 36.3%\n", "\u0411\u0430\u0441\u043a\u0435\u0442\u0431\u043e\u043b MC P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, 1 \u043c\u0430\u0442\u0447) = 84.8% P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, \u0441\u0435\u0440\u0438\u044f \u0434\u043e 4 \u043f\u043e\u0431\u0435\u0434) = 98.7% P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440, \u0441\u0435\u0440\u0438\u044f) = 1.3%\n" ] } ], "source": [ "import random, math\n", "from math import comb\n", "\n", "random.seed(42)\n", "\n", "def poisson_sample(lam):\n", " L, k, p = math.exp(-lam), 0, 1.0\n", " while True:\n", " k += 1\n", " p *= random.random()\n", " if p <= L:\n", " return k - 1\n", "\n", "def simulate(lam_fav, lam_dog, n=500_000):\n", " fav_wins = ties = 0\n", " for _ in range(n):\n", " f, d = poisson_sample(lam_fav), poisson_sample(lam_dog)\n", " if f > d:\n", " fav_wins += 1\n", " elif f == d:\n", " ties += 1\n", " return fav_wins / n, ties / n\n", "\n", "def series_win_prob(p, best_of=7):\n", " need = best_of // 2 + 1\n", " return sum(comb(need - 1 + l, l) * p**need * (1 - p)**l for l in range(need))\n", "\n", "for sport, lf, ld in [('\u0425\u043e\u043a\u043a\u0435\u0439', lam_fav_h, lam_dog_h),\n", " ('\u0411\u0430\u0441\u043a\u0435\u0442\u0431\u043e\u043b', lam_fav_b, lam_dog_b)]:\n", " win, tie = simulate(lf, ld)\n", " p_game = win + tie / 2\n", " p_series = series_win_prob(p_game)\n", " print(f'{sport:10s} MC P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, 1 \u043c\u0430\u0442\u0447) = {p_game:.1%} '\n", " f'P(\u0444\u0430\u0432\u043e\u0440\u0438\u0442, \u0441\u0435\u0440\u0438\u044f \u0434\u043e 4 \u043f\u043e\u0431\u0435\u0434) = {p_series:.1%} '\n", " f'P(\u0430\u0443\u0442\u0441\u0430\u0439\u0434\u0435\u0440, \u0441\u0435\u0440\u0438\u044f) = {1 - p_series:.1%}')" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.2" } }, "nbformat": 4, "nbformat_minor": 5 }