用Python做數學實驗

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?

先用已經知道的數學方法進行求解，假設贏一把的機率是P,求出P即可算出收益的期望。假設4次搖出的骰子正面分別為：x,y,z,w

則勝利的條件是：x+y+z+w<9,其中1<=x,y,z,w<=6

x+y+z+w = 9 時，有C_{8}^{3}種組合；

x+y+z+w = 8 時，有C_{7}^{3}種組合；

x+y+z+w = 7 時，有C_{6}^{3}種組合；

…

x+y+z+w = 4 時，有1種組合；

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

所以期望為： 7/72*9+65/72*(-1) = –1/36

用Python進行Monte Carlo 模擬，

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

Python實驗的結果則是：-0.47

雖然都是輸，但是結果有所出入