用Python做数学实验

Determine if you win or loose a hazard game.
Somebody suggests the following game. You pay 1 unit of money and are
allowed to throw four dice. If the sum of the eyes on the dice is less than 9,
you win 10 units of money, otherwise you loose your investment. Should you
play this game?

先用已经知道的数学方法进行求解，假设赢一把的概率是P,求出P即可算出收益的期望。假设4次摇出的骰子正面分别为：x,y,z,w

则胜利的条件是：x+y+z+w<9,其中1<=x,y,z,w<=6

x+y+z+w = 9 时，有C_{8}^{3}种组合；

x+y+z+w = 8 时，有C_{7}^{3}种组合；

x+y+z+w = 7 时，有C_{6}^{3}种组合；

…

x+y+z+w = 4 时，有1种组合；

所以，P = (C_{8}^{3}+C_{7}^{3}+C_{6}^{3}+C_{5}^{3}+C_{4}^{3}+C_{3}^{3})/64 = 7/72

所以期望为： 7/72*9+65/72*(-1) = –1/36

用Python进行Monte Carlo 仿真，

import random
n = 10000
m = 10000
money = 0
while(n):
b = []
for i in range(1,5):
b.append(random.randint(1,6))
if(sum(b)<9):
money+=9
else:
money-=1
n-=1
money = float(money) / m
print money

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Python实验的结果则是：-0.47

虽然都是输，但是结果有所出入