数概率

【Puzzle】向正方形区域扔硬币

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摘要: 《概率50题》向正方形投掷硬币

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参考: 《Fifty challenging problems in probability with solutions》

问题描述

Coin-in-Square

In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares

If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back

Otherwise, he loses the penny
If the penny lands on the table, what is his chance to win?

在一个游戏中，一个玩家从5英尺远的地方把一分钱硬币扔到一张1英寸正方形的桌子上，硬币直径为 3/4 英寸。

如果硬币完全落在正方形内，则玩家赢。

如果硬币落在桌上，他赢的机会有多大？

思路参考

如果硬币可以落在桌子上，那么硬币的重心在正方形范围内。

如果硬币完全落在正方形内，硬币重心到正方形四条边的距离都要大于硬币的半径。

硬币半径为 3/8 英寸。设硬币重心为 (x, y)，那么要满足以上条件，需要

求出 (x, y) 的范围是 (3/8 ~ 5/8, 3/8 ~ 5/8)。

因此当硬币落在桌子上的条件下，赢的机会有 (1/4 * 1/4) / 1 = 1/16 = 0.0625

蒙特卡洛模拟

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import numpy as np

def check(x, y):
    return x >= 3/8 and (1 - x) >= 3/8 and y >= 3/8 and (1 - y) >= 3/8

def test():
    T = int(1e7)
    s = 0
    for t in range(T):
        x = np.random.rand()
        y = np.random.rand()
        if check(x, y):
            s += 1
    print("his chance to win: {:.6f}".format(s / T))

for i in range(10):
    test()

模拟结果

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his chance to win: 0.062507
his chance to win: 0.062518
his chance to win: 0.062433
his chance to win: 0.062596
his chance to win: 0.062261
his chance to win: 0.062555
his chance to win: 0.062552
his chance to win: 0.062652
his chance to win: 0.062406
his chance to win: 0.062344
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