de Méré's paradox¶
https://carnotcycle.files.wordpress.com/2017/11/mere01.jpg?w=614&h=458
Chevalier de Méré (Antoine Gombaud 1607-1684) was a big spender in gambling circles. He found that it is worth to bet on the event of getting at least one 6 in four rolls of a die. He thought (misleaded by wrong calculation) that rolling two dice 24 times with a bet on having at least one double 6 has the same winning chance. But he found by experimenting that it is not true.
He wrote in 1654 to Blaise Pascal (1623-1662), who in turn shared the news of “De Méré’s paradox” with Pierre Fermat (1607-1665). They solved the “paradox” and with this the theory of probability was born.
Simulation with 4 dice (relative frequency)¶
Calculate the relative frequency of at least one 6 in four rolls of a die!
In [1]:
from random import randint
In [2]:
seq = [randint(1,6) for i in range(4)]
In [3]:
seq
Out[3]:
[6, 2, 5, 6]
Run the next code several times! Does it seem true that the relative frequency is more times greater than 0.5 than less, i.e. that it is worth betting that there will be at least one 6 among the 4 dice rolls?
In [4]:
n = 100
count = 0
for _ in range(n):
seq = [randint(1,6) for _ in range(4)]
if 6 in seq:
count += 1
print(count/n)
0.51
Probability with 4 dice¶
Calculate the exact probability of at least one 6 in four rolls of a die!
In [5]:
lst = [[i1, i2, i3, i4] for i1 in range(1,7) for i2 in range(1,7) \
for i3 in range(1,7) for i4 in range(1,7)]
count = 0
for i in lst:
if 6 in i:
count += 1
print(count, 6**4, count/6**4)
671 1296 0.5177469135802469
The next is almost the same code but there is a big difference in running. Instead of list comprehension use generator expression, as it does not construct a big list. It generates the elements when needed, that is in the for loop.
In [6]:
gen = ([i1, i2, i3, i4] for i1 in range(1,7) for i2 in range(1,7) \
for i3 in range(1,7) for i4 in range(1,7))
count = 0
for i in gen:
if 6 in i:
count += 1
print(count, 6**4, count/6**4)
671 1296 0.5177469135802469
Compair lst and gen:
In [7]:
lst # this is a long list of 1296 elements
Out[7]:
[[1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 1, 4], [1, 1, 1, 5], [1, 1, 1, 6], [1, 1, 2, 1], [1, 1, 2, 2], [1, 1, 2, 3], [1, 1, 2, 4], [1, 1, 2, 5], [1, 1, 2, 6], [1, 1, 3, 1], [1, 1, 3, 2], [1, 1, 3, 3], [1, 1, 3, 4], [1, 1, 3, 5], [1, 1, 3, 6], [1, 1, 4, 1], [1, 1, 4, 2], [1, 1, 4, 3], [1, 1, 4, 4], [1, 1, 4, 5], [1, 1, 4, 6], [1, 1, 5, 1], [1, 1, 5, 2], [1, 1, 5, 3], [1, 1, 5, 4], [1, 1, 5, 5], [1, 1, 5, 6], [1, 1, 6, 1], [1, 1, 6, 2], [1, 1, 6, 3], [1, 1, 6, 4], [1, 1, 6, 5], [1, 1, 6, 6], [1, 2, 1, 1], [1, 2, 1, 2], [1, 2, 1, 3], [1, 2, 1, 4], [1, 2, 1, 5], [1, 2, 1, 6], [1, 2, 2, 1], [1, 2, 2, 2], [1, 2, 2, 3], [1, 2, 2, 4], [1, 2, 2, 5], [1, 2, 2, 6], [1, 2, 3, 1], [1, 2, 3, 2], [1, 2, 3, 3], [1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 3, 6], [1, 2, 4, 1], [1, 2, 4, 2], [1, 2, 4, 3], [1, 2, 4, 4], [1, 2, 4, 5], [1, 2, 4, 6], [1, 2, 5, 1], [1, 2, 5, 2], [1, 2, 5, 3], [1, 2, 5, 4], [1, 2, 5, 5], [1, 2, 5, 6], [1, 2, 6, 1], [1, 2, 6, 2], [1, 2, 6, 3], [1, 2, 6, 4], [1, 2, 6, 5], [1, 2, 6, 6], [1, 3, 1, 1], [1, 3, 1, 2], [1, 3, 1, 3], [1, 3, 1, 4], [1, 3, 1, 5], [1, 3, 1, 6], [1, 3, 2, 1], [1, 3, 2, 2], [1, 3, 2, 3], [1, 3, 2, 4], [1, 3, 2, 5], [1, 3, 2, 6], [1, 3, 3, 1], [1, 3, 3, 2], [1, 3, 3, 3], [1, 3, 3, 4], [1, 3, 3, 5], [1, 3, 3, 6], [1, 3, 4, 1], [1, 3, 4, 2], [1, 3, 4, 3], [1, 3, 4, 4], [1, 3, 4, 5], [1, 3, 4, 6], [1, 3, 5, 1], [1, 3, 5, 2], [1, 3, 5, 3], [1, 3, 5, 4], [1, 3, 5, 5], [1, 3, 5, 6], [1, 3, 6, 1], [1, 3, 6, 2], [1, 3, 6, 3], [1, 3, 6, 4], [1, 3, 6, 5], [1, 3, 6, 6], [1, 4, 1, 1], [1, 4, 1, 2], [1, 4, 1, 3], [1, 4, 1, 4], [1, 4, 1, 5], [1, 4, 1, 6], [1, 4, 2, 1], [1, 4, 2, 2], [1, 4, 2, 3], [1, 4, 2, 4], [1, 4, 2, 5], [1, 4, 2, 6], [1, 4, 3, 1], [1, 4, 3, 2], [1, 4, 3, 3], [1, 4, 3, 4], [1, 4, 3, 5], [1, 4, 3, 6], [1, 4, 4, 1], [1, 4, 4, 2], [1, 4, 4, 3], [1, 4, 4, 4], [1, 4, 4, 5], [1, 4, 4, 6], [1, 4, 5, 1], [1, 4, 5, 2], [1, 4, 5, 3], [1, 4, 5, 4], [1, 4, 5, 5], [1, 4, 5, 6], [1, 4, 6, 1], [1, 4, 6, 2], [1, 4, 6, 3], [1, 4, 6, 4], [1, 4, 6, 5], [1, 4, 6, 6], [1, 5, 1, 1], [1, 5, 1, 2], [1, 5, 1, 3], [1, 5, 1, 4], [1, 5, 1, 5], [1, 5, 1, 6], [1, 5, 2, 1], [1, 5, 2, 2], [1, 5, 2, 3], [1, 5, 2, 4], [1, 5, 2, 5], [1, 5, 2, 6], [1, 5, 3, 1], [1, 5, 3, 2], [1, 5, 3, 3], [1, 5, 3, 4], [1, 5, 3, 5], [1, 5, 3, 6], [1, 5, 4, 1], [1, 5, 4, 2], [1, 5, 4, 3], [1, 5, 4, 4], [1, 5, 4, 5], [1, 5, 4, 6], [1, 5, 5, 1], [1, 5, 5, 2], [1, 5, 5, 3], [1, 5, 5, 4], [1, 5, 5, 5], [1, 5, 5, 6], [1, 5, 6, 1], [1, 5, 6, 2], [1, 5, 6, 3], [1, 5, 6, 4], [1, 5, 6, 5], [1, 5, 6, 6], [1, 6, 1, 1], [1, 6, 1, 2], [1, 6, 1, 3], [1, 6, 1, 4], [1, 6, 1, 5], [1, 6, 1, 6], [1, 6, 2, 1], [1, 6, 2, 2], [1, 6, 2, 3], [1, 6, 2, 4], [1, 6, 2, 5], [1, 6, 2, 6], [1, 6, 3, 1], [1, 6, 3, 2], [1, 6, 3, 3], [1, 6, 3, 4], [1, 6, 3, 5], [1, 6, 3, 6], [1, 6, 4, 1], [1, 6, 4, 2], [1, 6, 4, 3], [1, 6, 4, 4], [1, 6, 4, 5], [1, 6, 4, 6], [1, 6, 5, 1], [1, 6, 5, 2], [1, 6, 5, 3], [1, 6, 5, 4], [1, 6, 5, 5], [1, 6, 5, 6], [1, 6, 6, 1], [1, 6, 6, 2], [1, 6, 6, 3], [1, 6, 6, 4], [1, 6, 6, 5], [1, 6, 6, 6], [2, 1, 1, 1], [2, 1, 1, 2], [2, 1, 1, 3], [2, 1, 1, 4], [2, 1, 1, 5], [2, 1, 1, 6], [2, 1, 2, 1], [2, 1, 2, 2], [2, 1, 2, 3], [2, 1, 2, 4], [2, 1, 2, 5], [2, 1, 2, 6], [2, 1, 3, 1], [2, 1, 3, 2], [2, 1, 3, 3], [2, 1, 3, 4], [2, 1, 3, 5], [2, 1, 3, 6], [2, 1, 4, 1], [2, 1, 4, 2], [2, 1, 4, 3], [2, 1, 4, 4], [2, 1, 4, 5], [2, 1, 4, 6], [2, 1, 5, 1], [2, 1, 5, 2], [2, 1, 5, 3], [2, 1, 5, 4], [2, 1, 5, 5], [2, 1, 5, 6], [2, 1, 6, 1], [2, 1, 6, 2], [2, 1, 6, 3], [2, 1, 6, 4], [2, 1, 6, 5], [2, 1, 6, 6], [2, 2, 1, 1], [2, 2, 1, 2], [2, 2, 1, 3], [2, 2, 1, 4], [2, 2, 1, 5], [2, 2, 1, 6], [2, 2, 2, 1], [2, 2, 2, 2], [2, 2, 2, 3], [2, 2, 2, 4], [2, 2, 2, 5], [2, 2, 2, 6], [2, 2, 3, 1], [2, 2, 3, 2], [2, 2, 3, 3], [2, 2, 3, 4], [2, 2, 3, 5], [2, 2, 3, 6], [2, 2, 4, 1], [2, 2, 4, 2], [2, 2, 4, 3], [2, 2, 4, 4], [2, 2, 4, 5], [2, 2, 4, 6], [2, 2, 5, 1], [2, 2, 5, 2], [2, 2, 5, 3], [2, 2, 5, 4], [2, 2, 5, 5], [2, 2, 5, 6], [2, 2, 6, 1], [2, 2, 6, 2], [2, 2, 6, 3], [2, 2, 6, 4], [2, 2, 6, 5], [2, 2, 6, 6], [2, 3, 1, 1], [2, 3, 1, 2], [2, 3, 1, 3], [2, 3, 1, 4], [2, 3, 1, 5], [2, 3, 1, 6], [2, 3, 2, 1], [2, 3, 2, 2], [2, 3, 2, 3], [2, 3, 2, 4], [2, 3, 2, 5], [2, 3, 2, 6], [2, 3, 3, 1], [2, 3, 3, 2], [2, 3, 3, 3], [2, 3, 3, 4], [2, 3, 3, 5], [2, 3, 3, 6], [2, 3, 4, 1], [2, 3, 4, 2], [2, 3, 4, 3], [2, 3, 4, 4], [2, 3, 4, 5], [2, 3, 4, 6], [2, 3, 5, 1], [2, 3, 5, 2], [2, 3, 5, 3], [2, 3, 5, 4], [2, 3, 5, 5], [2, 3, 5, 6], [2, 3, 6, 1], [2, 3, 6, 2], [2, 3, 6, 3], [2, 3, 6, 4], [2, 3, 6, 5], [2, 3, 6, 6], [2, 4, 1, 1], [2, 4, 1, 2], [2, 4, 1, 3], [2, 4, 1, 4], [2, 4, 1, 5], [2, 4, 1, 6], [2, 4, 2, 1], [2, 4, 2, 2], [2, 4, 2, 3], [2, 4, 2, 4], [2, 4, 2, 5], [2, 4, 2, 6], [2, 4, 3, 1], [2, 4, 3, 2], [2, 4, 3, 3], [2, 4, 3, 4], [2, 4, 3, 5], [2, 4, 3, 6], [2, 4, 4, 1], [2, 4, 4, 2], [2, 4, 4, 3], [2, 4, 4, 4], [2, 4, 4, 5], [2, 4, 4, 6], [2, 4, 5, 1], [2, 4, 5, 2], [2, 4, 5, 3], [2, 4, 5, 4], [2, 4, 5, 5], [2, 4, 5, 6], [2, 4, 6, 1], [2, 4, 6, 2], [2, 4, 6, 3], [2, 4, 6, 4], [2, 4, 6, 5], [2, 4, 6, 6], [2, 5, 1, 1], [2, 5, 1, 2], [2, 5, 1, 3], [2, 5, 1, 4], [2, 5, 1, 5], [2, 5, 1, 6], [2, 5, 2, 1], [2, 5, 2, 2], [2, 5, 2, 3], [2, 5, 2, 4], [2, 5, 2, 5], [2, 5, 2, 6], [2, 5, 3, 1], [2, 5, 3, 2], [2, 5, 3, 3], [2, 5, 3, 4], [2, 5, 3, 5], [2, 5, 3, 6], [2, 5, 4, 1], [2, 5, 4, 2], [2, 5, 4, 3], [2, 5, 4, 4], [2, 5, 4, 5], [2, 5, 4, 6], [2, 5, 5, 1], [2, 5, 5, 2], [2, 5, 5, 3], [2, 5, 5, 4], [2, 5, 5, 5], [2, 5, 5, 6], [2, 5, 6, 1], [2, 5, 6, 2], [2, 5, 6, 3], [2, 5, 6, 4], [2, 5, 6, 5], [2, 5, 6, 6], [2, 6, 1, 1], [2, 6, 1, 2], [2, 6, 1, 3], [2, 6, 1, 4], [2, 6, 1, 5], [2, 6, 1, 6], [2, 6, 2, 1], [2, 6, 2, 2], [2, 6, 2, 3], [2, 6, 2, 4], [2, 6, 2, 5], [2, 6, 2, 6], [2, 6, 3, 1], [2, 6, 3, 2], [2, 6, 3, 3], [2, 6, 3, 4], [2, 6, 3, 5], [2, 6, 3, 6], [2, 6, 4, 1], [2, 6, 4, 2], [2, 6, 4, 3], [2, 6, 4, 4], [2, 6, 4, 5], [2, 6, 4, 6], [2, 6, 5, 1], [2, 6, 5, 2], [2, 6, 5, 3], [2, 6, 5, 4], [2, 6, 5, 5], [2, 6, 5, 6], [2, 6, 6, 1], [2, 6, 6, 2], [2, 6, 6, 3], [2, 6, 6, 4], [2, 6, 6, 5], [2, 6, 6, 6], [3, 1, 1, 1], [3, 1, 1, 2], [3, 1, 1, 3], [3, 1, 1, 4], [3, 1, 1, 5], [3, 1, 1, 6], [3, 1, 2, 1], [3, 1, 2, 2], [3, 1, 2, 3], [3, 1, 2, 4], [3, 1, 2, 5], [3, 1, 2, 6], [3, 1, 3, 1], [3, 1, 3, 2], [3, 1, 3, 3], [3, 1, 3, 4], [3, 1, 3, 5], [3, 1, 3, 6], [3, 1, 4, 1], [3, 1, 4, 2], [3, 1, 4, 3], [3, 1, 4, 4], [3, 1, 4, 5], [3, 1, 4, 6], [3, 1, 5, 1], [3, 1, 5, 2], [3, 1, 5, 3], [3, 1, 5, 4], [3, 1, 5, 5], [3, 1, 5, 6], [3, 1, 6, 1], [3, 1, 6, 2], [3, 1, 6, 3], [3, 1, 6, 4], [3, 1, 6, 5], [3, 1, 6, 6], [3, 2, 1, 1], [3, 2, 1, 2], [3, 2, 1, 3], [3, 2, 1, 4], [3, 2, 1, 5], [3, 2, 1, 6], [3, 2, 2, 1], [3, 2, 2, 2], [3, 2, 2, 3], [3, 2, 2, 4], [3, 2, 2, 5], [3, 2, 2, 6], [3, 2, 3, 1], [3, 2, 3, 2], [3, 2, 3, 3], [3, 2, 3, 4], [3, 2, 3, 5], [3, 2, 3, 6], [3, 2, 4, 1], [3, 2, 4, 2], [3, 2, 4, 3], [3, 2, 4, 4], [3, 2, 4, 5], [3, 2, 4, 6], [3, 2, 5, 1], [3, 2, 5, 2], [3, 2, 5, 3], [3, 2, 5, 4], [3, 2, 5, 5], [3, 2, 5, 6], [3, 2, 6, 1], [3, 2, 6, 2], [3, 2, 6, 3], [3, 2, 6, 4], [3, 2, 6, 5], [3, 2, 6, 6], [3, 3, 1, 1], [3, 3, 1, 2], [3, 3, 1, 3], [3, 3, 1, 4], [3, 3, 1, 5], [3, 3, 1, 6], [3, 3, 2, 1], [3, 3, 2, 2], [3, 3, 2, 3], [3, 3, 2, 4], [3, 3, 2, 5], [3, 3, 2, 6], [3, 3, 3, 1], [3, 3, 3, 2], [3, 3, 3, 3], [3, 3, 3, 4], [3, 3, 3, 5], [3, 3, 3, 6], [3, 3, 4, 1], [3, 3, 4, 2], [3, 3, 4, 3], [3, 3, 4, 4], [3, 3, 4, 5], [3, 3, 4, 6], [3, 3, 5, 1], [3, 3, 5, 2], [3, 3, 5, 3], [3, 3, 5, 4], [3, 3, 5, 5], [3, 3, 5, 6], [3, 3, 6, 1], [3, 3, 6, 2], [3, 3, 6, 3], [3, 3, 6, 4], [3, 3, 6, 5], [3, 3, 6, 6], [3, 4, 1, 1], [3, 4, 1, 2], [3, 4, 1, 3], [3, 4, 1, 4], [3, 4, 1, 5], [3, 4, 1, 6], [3, 4, 2, 1], [3, 4, 2, 2], [3, 4, 2, 3], [3, 4, 2, 4], [3, 4, 2, 5], [3, 4, 2, 6], [3, 4, 3, 1], [3, 4, 3, 2], [3, 4, 3, 3], [3, 4, 3, 4], [3, 4, 3, 5], [3, 4, 3, 6], [3, 4, 4, 1], [3, 4, 4, 2], [3, 4, 4, 3], [3, 4, 4, 4], [3, 4, 4, 5], [3, 4, 4, 6], [3, 4, 5, 1], [3, 4, 5, 2], [3, 4, 5, 3], [3, 4, 5, 4], [3, 4, 5, 5], [3, 4, 5, 6], [3, 4, 6, 1], [3, 4, 6, 2], [3, 4, 6, 3], [3, 4, 6, 4], [3, 4, 6, 5], [3, 4, 6, 6], [3, 5, 1, 1], [3, 5, 1, 2], [3, 5, 1, 3], [3, 5, 1, 4], [3, 5, 1, 5], [3, 5, 1, 6], [3, 5, 2, 1], [3, 5, 2, 2], [3, 5, 2, 3], [3, 5, 2, 4], [3, 5, 2, 5], [3, 5, 2, 6], [3, 5, 3, 1], [3, 5, 3, 2], [3, 5, 3, 3], [3, 5, 3, 4], [3, 5, 3, 5], [3, 5, 3, 6], [3, 5, 4, 1], [3, 5, 4, 2], [3, 5, 4, 3], [3, 5, 4, 4], [3, 5, 4, 5], [3, 5, 4, 6], [3, 5, 5, 1], [3, 5, 5, 2], [3, 5, 5, 3], [3, 5, 5, 4], [3, 5, 5, 5], [3, 5, 5, 6], [3, 5, 6, 1], [3, 5, 6, 2], [3, 5, 6, 3], [3, 5, 6, 4], [3, 5, 6, 5], [3, 5, 6, 6], [3, 6, 1, 1], [3, 6, 1, 2], [3, 6, 1, 3], [3, 6, 1, 4], [3, 6, 1, 5], [3, 6, 1, 6], [3, 6, 2, 1], [3, 6, 2, 2], [3, 6, 2, 3], [3, 6, 2, 4], [3, 6, 2, 5], [3, 6, 2, 6], [3, 6, 3, 1], [3, 6, 3, 2], [3, 6, 3, 3], [3, 6, 3, 4], [3, 6, 3, 5], [3, 6, 3, 6], [3, 6, 4, 1], [3, 6, 4, 2], [3, 6, 4, 3], [3, 6, 4, 4], [3, 6, 4, 5], [3, 6, 4, 6], [3, 6, 5, 1], [3, 6, 5, 2], [3, 6, 5, 3], [3, 6, 5, 4], [3, 6, 5, 5], [3, 6, 5, 6], [3, 6, 6, 1], [3, 6, 6, 2], [3, 6, 6, 3], [3, 6, 6, 4], [3, 6, 6, 5], [3, 6, 6, 6], [4, 1, 1, 1], [4, 1, 1, 2], [4, 1, 1, 3], [4, 1, 1, 4], [4, 1, 1, 5], [4, 1, 1, 6], [4, 1, 2, 1], [4, 1, 2, 2], [4, 1, 2, 3], [4, 1, 2, 4], [4, 1, 2, 5], [4, 1, 2, 6], [4, 1, 3, 1], [4, 1, 3, 2], [4, 1, 3, 3], [4, 1, 3, 4], [4, 1, 3, 5], [4, 1, 3, 6], [4, 1, 4, 1], [4, 1, 4, 2], [4, 1, 4, 3], [4, 1, 4, 4], [4, 1, 4, 5], [4, 1, 4, 6], [4, 1, 5, 1], [4, 1, 5, 2], [4, 1, 5, 3], [4, 1, 5, 4], [4, 1, 5, 5], [4, 1, 5, 6], [4, 1, 6, 1], [4, 1, 6, 2], [4, 1, 6, 3], [4, 1, 6, 4], [4, 1, 6, 5], [4, 1, 6, 6], [4, 2, 1, 1], [4, 2, 1, 2], [4, 2, 1, 3], [4, 2, 1, 4], [4, 2, 1, 5], [4, 2, 1, 6], [4, 2, 2, 1], [4, 2, 2, 2], [4, 2, 2, 3], [4, 2, 2, 4], [4, 2, 2, 5], [4, 2, 2, 6], [4, 2, 3, 1], [4, 2, 3, 2], [4, 2, 3, 3], [4, 2, 3, 4], [4, 2, 3, 5], [4, 2, 3, 6], [4, 2, 4, 1], [4, 2, 4, 2], [4, 2, 4, 3], [4, 2, 4, 4], [4, 2, 4, 5], [4, 2, 4, 6], [4, 2, 5, 1], [4, 2, 5, 2], [4, 2, 5, 3], [4, 2, 5, 4], [4, 2, 5, 5], [4, 2, 5, 6], [4, 2, 6, 1], [4, 2, 6, 2], [4, 2, 6, 3], [4, 2, 6, 4], [4, 2, 6, 5], [4, 2, 6, 6], [4, 3, 1, 1], [4, 3, 1, 2], [4, 3, 1, 3], [4, 3, 1, 4], [4, 3, 1, 5], [4, 3, 1, 6], [4, 3, 2, 1], [4, 3, 2, 2], [4, 3, 2, 3], [4, 3, 2, 4], [4, 3, 2, 5], [4, 3, 2, 6], [4, 3, 3, 1], [4, 3, 3, 2], [4, 3, 3, 3], [4, 3, 3, 4], [4, 3, 3, 5], [4, 3, 3, 6], [4, 3, 4, 1], [4, 3, 4, 2], [4, 3, 4, 3], [4, 3, 4, 4], [4, 3, 4, 5], [4, 3, 4, 6], [4, 3, 5, 1], [4, 3, 5, 2], [4, 3, 5, 3], [4, 3, 5, 4], [4, 3, 5, 5], [4, 3, 5, 6], [4, 3, 6, 1], [4, 3, 6, 2], [4, 3, 6, 3], [4, 3, 6, 4], [4, 3, 6, 5], [4, 3, 6, 6], [4, 4, 1, 1], [4, 4, 1, 2], [4, 4, 1, 3], [4, 4, 1, 4], [4, 4, 1, 5], [4, 4, 1, 6], [4, 4, 2, 1], [4, 4, 2, 2], [4, 4, 2, 3], [4, 4, 2, 4], [4, 4, 2, 5], [4, 4, 2, 6], [4, 4, 3, 1], [4, 4, 3, 2], [4, 4, 3, 3], [4, 4, 3, 4], [4, 4, 3, 5], [4, 4, 3, 6], [4, 4, 4, 1], [4, 4, 4, 2], [4, 4, 4, 3], [4, 4, 4, 4], [4, 4, 4, 5], [4, 4, 4, 6], [4, 4, 5, 1], [4, 4, 5, 2], [4, 4, 5, 3], [4, 4, 5, 4], [4, 4, 5, 5], [4, 4, 5, 6], [4, 4, 6, 1], [4, 4, 6, 2], [4, 4, 6, 3], [4, 4, 6, 4], [4, 4, 6, 5], [4, 4, 6, 6], [4, 5, 1, 1], [4, 5, 1, 2], [4, 5, 1, 3], [4, 5, 1, 4], [4, 5, 1, 5], [4, 5, 1, 6], [4, 5, 2, 1], [4, 5, 2, 2], [4, 5, 2, 3], [4, 5, 2, 4], [4, 5, 2, 5], [4, 5, 2, 6], [4, 5, 3, 1], [4, 5, 3, 2], [4, 5, 3, 3], [4, 5, 3, 4], [4, 5, 3, 5], [4, 5, 3, 6], [4, 5, 4, 1], [4, 5, 4, 2], [4, 5, 4, 3], [4, 5, 4, 4], [4, 5, 4, 5], [4, 5, 4, 6], [4, 5, 5, 1], [4, 5, 5, 2], [4, 5, 5, 3], [4, 5, 5, 4], [4, 5, 5, 5], [4, 5, 5, 6], [4, 5, 6, 1], [4, 5, 6, 2], [4, 5, 6, 3], [4, 5, 6, 4], [4, 5, 6, 5], [4, 5, 6, 6], [4, 6, 1, 1], [4, 6, 1, 2], [4, 6, 1, 3], [4, 6, 1, 4], [4, 6, 1, 5], [4, 6, 1, 6], [4, 6, 2, 1], [4, 6, 2, 2], [4, 6, 2, 3], [4, 6, 2, 4], [4, 6, 2, 5], [4, 6, 2, 6], [4, 6, 3, 1], [4, 6, 3, 2], [4, 6, 3, 3], [4, 6, 3, 4], [4, 6, 3, 5], [4, 6, 3, 6], [4, 6, 4, 1], [4, 6, 4, 2], [4, 6, 4, 3], [4, 6, 4, 4], [4, 6, 4, 5], [4, 6, 4, 6], [4, 6, 5, 1], [4, 6, 5, 2], [4, 6, 5, 3], [4, 6, 5, 4], [4, 6, 5, 5], [4, 6, 5, 6], [4, 6, 6, 1], [4, 6, 6, 2], [4, 6, 6, 3], [4, 6, 6, 4], [4, 6, 6, 5], [4, 6, 6, 6], [5, 1, 1, 1], [5, 1, 1, 2], [5, 1, 1, 3], [5, 1, 1, 4], [5, 1, 1, 5], [5, 1, 1, 6], [5, 1, 2, 1], [5, 1, 2, 2], [5, 1, 2, 3], [5, 1, 2, 4], [5, 1, 2, 5], [5, 1, 2, 6], [5, 1, 3, 1], [5, 1, 3, 2], [5, 1, 3, 3], [5, 1, 3, 4], [5, 1, 3, 5], [5, 1, 3, 6], [5, 1, 4, 1], [5, 1, 4, 2], [5, 1, 4, 3], [5, 1, 4, 4], [5, 1, 4, 5], [5, 1, 4, 6], [5, 1, 5, 1], [5, 1, 5, 2], [5, 1, 5, 3], [5, 1, 5, 4], [5, 1, 5, 5], [5, 1, 5, 6], [5, 1, 6, 1], [5, 1, 6, 2], [5, 1, 6, 3], [5, 1, 6, 4], [5, 1, 6, 5], [5, 1, 6, 6], [5, 2, 1, 1], [5, 2, 1, 2], [5, 2, 1, 3], [5, 2, 1, 4], [5, 2, 1, 5], [5, 2, 1, 6], [5, 2, 2, 1], [5, 2, 2, 2], [5, 2, 2, 3], [5, 2, 2, 4], [5, 2, 2, 5], [5, 2, 2, 6], [5, 2, 3, 1], [5, 2, 3, 2], [5, 2, 3, 3], [5, 2, 3, 4], [5, 2, 3, 5], [5, 2, 3, 6], [5, 2, 4, 1], [5, 2, 4, 2], [5, 2, 4, 3], [5, 2, 4, 4], [5, 2, 4, 5], [5, 2, 4, 6], [5, 2, 5, 1], [5, 2, 5, 2], [5, 2, 5, 3], [5, 2, 5, 4], [5, 2, 5, 5], [5, 2, 5, 6], [5, 2, 6, 1], [5, 2, 6, 2], [5, 2, 6, 3], [5, 2, 6, 4], [5, 2, 6, 5], [5, 2, 6, 6], [5, 3, 1, 1], [5, 3, 1, 2], [5, 3, 1, 3], [5, 3, 1, 4], [5, 3, 1, 5], [5, 3, 1, 6], [5, 3, 2, 1], [5, 3, 2, 2], [5, 3, 2, 3], [5, 3, 2, 4], [5, 3, 2, 5], [5, 3, 2, 6], [5, 3, 3, 1], [5, 3, 3, 2], [5, 3, 3, 3], [5, 3, 3, 4], [5, 3, 3, 5], [5, 3, 3, 6], [5, 3, 4, 1], [5, 3, 4, 2], [5, 3, 4, 3], [5, 3, 4, 4], [5, 3, 4, 5], [5, 3, 4, 6], [5, 3, 5, 1], [5, 3, 5, 2], [5, 3, 5, 3], [5, 3, 5, 4], [5, 3, 5, 5], [5, 3, 5, 6], [5, 3, 6, 1], [5, 3, 6, 2], [5, 3, 6, 3], [5, 3, 6, 4], [5, 3, 6, 5], [5, 3, 6, 6], [5, 4, 1, 1], [5, 4, 1, 2], [5, 4, 1, 3], [5, 4, 1, 4], [5, 4, 1, 5], [5, 4, 1, 6], [5, 4, 2, 1], [5, 4, 2, 2], [5, 4, 2, 3], [5, 4, 2, 4], [5, 4, 2, 5], [5, 4, 2, 6], [5, 4, 3, 1], [5, 4, 3, 2], [5, 4, 3, 3], [5, 4, 3, 4], [5, 4, 3, 5], [5, 4, 3, 6], [5, 4, 4, 1], [5, 4, 4, 2], [5, 4, 4, 3], [5, 4, 4, 4], [5, 4, 4, 5], [5, 4, 4, 6], [5, 4, 5, 1], [5, 4, 5, 2], [5, 4, 5, 3], [5, 4, 5, 4], ...]
In [8]:
gen
Out[8]:
<generator object <genexpr> at 0x104519580>
Example: Write a complete program putting the codes above into functions (simulation and calculating probability) and call them in the main funcion! Save this code into a file and run it from terminal!
In [9]:
from random import randint
def simulate(n):
count = 0
for _ in range(n):
seq = [randint(1,6) for i in range(4)]
if 6 in seq:
count += 1
return count/n
def probability():
gen = ([i1, i2, i3, i4] for i1 in range(1,7) for i2 in range(1,7) \
for i3 in range(1,7) for i4 in range(1,7))
count = 0
for i in gen:
if 6 in i:
count += 1
return count/6**4
def main():
print(f" winning")
print(f"100 experiments {simulate(100):.5f}")
print(f"10000 experiments {simulate(10000):.5f}")
print(f"probability {probability():.5f}")
if __name__ == "__main__":
main()
winning 100 experiments 0.50000 10000 experiments 0.51690 probability 0.51775
Using a little mathematics, we could calculate the probability without listing all possibilities:
In [10]:
1-(5/6)**4
Out[10]:
0.5177469135802468
A programming tip: listing the results of several embedded loops we may use the product function of itertools module.
In [11]:
from itertools import product
dice = range(1, 7)
four = product(dice, dice, dice, dice)
for _ in four:
print(_)
(1, 1, 1, 1) (1, 1, 1, 2) (1, 1, 1, 3) (1, 1, 1, 4) (1, 1, 1, 5) (1, 1, 1, 6) (1, 1, 2, 1) (1, 1, 2, 2) (1, 1, 2, 3) (1, 1, 2, 4) (1, 1, 2, 5) (1, 1, 2, 6) (1, 1, 3, 1) (1, 1, 3, 2) (1, 1, 3, 3) (1, 1, 3, 4) (1, 1, 3, 5) (1, 1, 3, 6) (1, 1, 4, 1) (1, 1, 4, 2) (1, 1, 4, 3) (1, 1, 4, 4) (1, 1, 4, 5) (1, 1, 4, 6) (1, 1, 5, 1) (1, 1, 5, 2) (1, 1, 5, 3) (1, 1, 5, 4) (1, 1, 5, 5) (1, 1, 5, 6) (1, 1, 6, 1) (1, 1, 6, 2) (1, 1, 6, 3) (1, 1, 6, 4) (1, 1, 6, 5) (1, 1, 6, 6) (1, 2, 1, 1) (1, 2, 1, 2) (1, 2, 1, 3) (1, 2, 1, 4) (1, 2, 1, 5) (1, 2, 1, 6) (1, 2, 2, 1) (1, 2, 2, 2) (1, 2, 2, 3) (1, 2, 2, 4) (1, 2, 2, 5) (1, 2, 2, 6) (1, 2, 3, 1) (1, 2, 3, 2) (1, 2, 3, 3) (1, 2, 3, 4) (1, 2, 3, 5) (1, 2, 3, 6) (1, 2, 4, 1) (1, 2, 4, 2) (1, 2, 4, 3) (1, 2, 4, 4) (1, 2, 4, 5) (1, 2, 4, 6) (1, 2, 5, 1) (1, 2, 5, 2) (1, 2, 5, 3) (1, 2, 5, 4) (1, 2, 5, 5) (1, 2, 5, 6) (1, 2, 6, 1) (1, 2, 6, 2) (1, 2, 6, 3) (1, 2, 6, 4) (1, 2, 6, 5) (1, 2, 6, 6) (1, 3, 1, 1) (1, 3, 1, 2) (1, 3, 1, 3) (1, 3, 1, 4) (1, 3, 1, 5) (1, 3, 1, 6) (1, 3, 2, 1) (1, 3, 2, 2) (1, 3, 2, 3) (1, 3, 2, 4) (1, 3, 2, 5) (1, 3, 2, 6) (1, 3, 3, 1) (1, 3, 3, 2) (1, 3, 3, 3) (1, 3, 3, 4) (1, 3, 3, 5) (1, 3, 3, 6) (1, 3, 4, 1) (1, 3, 4, 2) (1, 3, 4, 3) (1, 3, 4, 4) (1, 3, 4, 5) (1, 3, 4, 6) (1, 3, 5, 1) (1, 3, 5, 2) (1, 3, 5, 3) (1, 3, 5, 4) (1, 3, 5, 5) (1, 3, 5, 6) (1, 3, 6, 1) (1, 3, 6, 2) (1, 3, 6, 3) (1, 3, 6, 4) (1, 3, 6, 5) (1, 3, 6, 6) (1, 4, 1, 1) (1, 4, 1, 2) (1, 4, 1, 3) (1, 4, 1, 4) (1, 4, 1, 5) (1, 4, 1, 6) (1, 4, 2, 1) (1, 4, 2, 2) (1, 4, 2, 3) (1, 4, 2, 4) (1, 4, 2, 5) (1, 4, 2, 6) (1, 4, 3, 1) (1, 4, 3, 2) (1, 4, 3, 3) (1, 4, 3, 4) (1, 4, 3, 5) (1, 4, 3, 6) (1, 4, 4, 1) (1, 4, 4, 2) (1, 4, 4, 3) (1, 4, 4, 4) (1, 4, 4, 5) (1, 4, 4, 6) (1, 4, 5, 1) (1, 4, 5, 2) (1, 4, 5, 3) (1, 4, 5, 4) (1, 4, 5, 5) (1, 4, 5, 6) (1, 4, 6, 1) (1, 4, 6, 2) (1, 4, 6, 3) (1, 4, 6, 4) (1, 4, 6, 5) (1, 4, 6, 6) (1, 5, 1, 1) (1, 5, 1, 2) (1, 5, 1, 3) (1, 5, 1, 4) (1, 5, 1, 5) (1, 5, 1, 6) (1, 5, 2, 1) (1, 5, 2, 2) (1, 5, 2, 3) (1, 5, 2, 4) (1, 5, 2, 5) (1, 5, 2, 6) (1, 5, 3, 1) (1, 5, 3, 2) (1, 5, 3, 3) (1, 5, 3, 4) (1, 5, 3, 5) (1, 5, 3, 6) (1, 5, 4, 1) (1, 5, 4, 2) (1, 5, 4, 3) (1, 5, 4, 4) (1, 5, 4, 5) (1, 5, 4, 6) (1, 5, 5, 1) (1, 5, 5, 2) (1, 5, 5, 3) (1, 5, 5, 4) (1, 5, 5, 5) (1, 5, 5, 6) (1, 5, 6, 1) (1, 5, 6, 2) (1, 5, 6, 3) (1, 5, 6, 4) (1, 5, 6, 5) (1, 5, 6, 6) (1, 6, 1, 1) (1, 6, 1, 2) (1, 6, 1, 3) (1, 6, 1, 4) (1, 6, 1, 5) (1, 6, 1, 6) (1, 6, 2, 1) (1, 6, 2, 2) (1, 6, 2, 3) (1, 6, 2, 4) (1, 6, 2, 5) (1, 6, 2, 6) (1, 6, 3, 1) (1, 6, 3, 2) (1, 6, 3, 3) (1, 6, 3, 4) (1, 6, 3, 5) (1, 6, 3, 6) (1, 6, 4, 1) (1, 6, 4, 2) (1, 6, 4, 3) (1, 6, 4, 4) (1, 6, 4, 5) (1, 6, 4, 6) (1, 6, 5, 1) (1, 6, 5, 2) (1, 6, 5, 3) (1, 6, 5, 4) (1, 6, 5, 5) (1, 6, 5, 6) (1, 6, 6, 1) (1, 6, 6, 2) (1, 6, 6, 3) (1, 6, 6, 4) (1, 6, 6, 5) (1, 6, 6, 6) (2, 1, 1, 1) (2, 1, 1, 2) (2, 1, 1, 3) (2, 1, 1, 4) (2, 1, 1, 5) (2, 1, 1, 6) (2, 1, 2, 1) (2, 1, 2, 2) (2, 1, 2, 3) (2, 1, 2, 4) (2, 1, 2, 5) (2, 1, 2, 6) (2, 1, 3, 1) (2, 1, 3, 2) (2, 1, 3, 3) (2, 1, 3, 4) (2, 1, 3, 5) (2, 1, 3, 6) (2, 1, 4, 1) (2, 1, 4, 2) (2, 1, 4, 3) (2, 1, 4, 4) (2, 1, 4, 5) (2, 1, 4, 6) (2, 1, 5, 1) (2, 1, 5, 2) (2, 1, 5, 3) (2, 1, 5, 4) (2, 1, 5, 5) (2, 1, 5, 6) (2, 1, 6, 1) (2, 1, 6, 2) (2, 1, 6, 3) (2, 1, 6, 4) (2, 1, 6, 5) (2, 1, 6, 6) (2, 2, 1, 1) (2, 2, 1, 2) (2, 2, 1, 3) (2, 2, 1, 4) (2, 2, 1, 5) (2, 2, 1, 6) (2, 2, 2, 1) (2, 2, 2, 2) (2, 2, 2, 3) (2, 2, 2, 4) (2, 2, 2, 5) (2, 2, 2, 6) (2, 2, 3, 1) (2, 2, 3, 2) (2, 2, 3, 3) (2, 2, 3, 4) (2, 2, 3, 5) (2, 2, 3, 6) (2, 2, 4, 1) (2, 2, 4, 2) (2, 2, 4, 3) (2, 2, 4, 4) (2, 2, 4, 5) (2, 2, 4, 6) (2, 2, 5, 1) (2, 2, 5, 2) (2, 2, 5, 3) (2, 2, 5, 4) (2, 2, 5, 5) (2, 2, 5, 6) (2, 2, 6, 1) (2, 2, 6, 2) (2, 2, 6, 3) (2, 2, 6, 4) (2, 2, 6, 5) (2, 2, 6, 6) (2, 3, 1, 1) (2, 3, 1, 2) (2, 3, 1, 3) (2, 3, 1, 4) (2, 3, 1, 5) (2, 3, 1, 6) (2, 3, 2, 1) (2, 3, 2, 2) (2, 3, 2, 3) (2, 3, 2, 4) (2, 3, 2, 5) (2, 3, 2, 6) (2, 3, 3, 1) (2, 3, 3, 2) (2, 3, 3, 3) (2, 3, 3, 4) (2, 3, 3, 5) (2, 3, 3, 6) (2, 3, 4, 1) (2, 3, 4, 2) (2, 3, 4, 3) (2, 3, 4, 4) (2, 3, 4, 5) (2, 3, 4, 6) (2, 3, 5, 1) (2, 3, 5, 2) (2, 3, 5, 3) (2, 3, 5, 4) (2, 3, 5, 5) (2, 3, 5, 6) (2, 3, 6, 1) (2, 3, 6, 2) (2, 3, 6, 3) (2, 3, 6, 4) (2, 3, 6, 5) (2, 3, 6, 6) (2, 4, 1, 1) (2, 4, 1, 2) (2, 4, 1, 3) (2, 4, 1, 4) (2, 4, 1, 5) (2, 4, 1, 6) (2, 4, 2, 1) (2, 4, 2, 2) (2, 4, 2, 3) (2, 4, 2, 4) (2, 4, 2, 5) (2, 4, 2, 6) (2, 4, 3, 1) (2, 4, 3, 2) (2, 4, 3, 3) (2, 4, 3, 4) (2, 4, 3, 5) (2, 4, 3, 6) (2, 4, 4, 1) (2, 4, 4, 2) (2, 4, 4, 3) (2, 4, 4, 4) (2, 4, 4, 5) (2, 4, 4, 6) (2, 4, 5, 1) (2, 4, 5, 2) (2, 4, 5, 3) (2, 4, 5, 4) (2, 4, 5, 5) (2, 4, 5, 6) (2, 4, 6, 1) (2, 4, 6, 2) (2, 4, 6, 3) (2, 4, 6, 4) (2, 4, 6, 5) (2, 4, 6, 6) (2, 5, 1, 1) (2, 5, 1, 2) (2, 5, 1, 3) (2, 5, 1, 4) (2, 5, 1, 5) (2, 5, 1, 6) (2, 5, 2, 1) (2, 5, 2, 2) (2, 5, 2, 3) (2, 5, 2, 4) (2, 5, 2, 5) (2, 5, 2, 6) (2, 5, 3, 1) (2, 5, 3, 2) (2, 5, 3, 3) (2, 5, 3, 4) (2, 5, 3, 5) (2, 5, 3, 6) (2, 5, 4, 1) (2, 5, 4, 2) (2, 5, 4, 3) (2, 5, 4, 4) (2, 5, 4, 5) (2, 5, 4, 6) (2, 5, 5, 1) (2, 5, 5, 2) (2, 5, 5, 3) (2, 5, 5, 4) (2, 5, 5, 5) (2, 5, 5, 6) (2, 5, 6, 1) (2, 5, 6, 2) (2, 5, 6, 3) (2, 5, 6, 4) (2, 5, 6, 5) (2, 5, 6, 6) (2, 6, 1, 1) (2, 6, 1, 2) (2, 6, 1, 3) (2, 6, 1, 4) (2, 6, 1, 5) (2, 6, 1, 6) (2, 6, 2, 1) (2, 6, 2, 2) (2, 6, 2, 3) (2, 6, 2, 4) (2, 6, 2, 5) (2, 6, 2, 6) (2, 6, 3, 1) (2, 6, 3, 2) (2, 6, 3, 3) (2, 6, 3, 4) (2, 6, 3, 5) (2, 6, 3, 6) (2, 6, 4, 1) (2, 6, 4, 2) (2, 6, 4, 3) (2, 6, 4, 4) (2, 6, 4, 5) (2, 6, 4, 6) (2, 6, 5, 1) (2, 6, 5, 2) (2, 6, 5, 3) (2, 6, 5, 4) (2, 6, 5, 5) (2, 6, 5, 6) (2, 6, 6, 1) (2, 6, 6, 2) (2, 6, 6, 3) (2, 6, 6, 4) (2, 6, 6, 5) (2, 6, 6, 6) (3, 1, 1, 1) (3, 1, 1, 2) (3, 1, 1, 3) (3, 1, 1, 4) (3, 1, 1, 5) (3, 1, 1, 6) (3, 1, 2, 1) (3, 1, 2, 2) (3, 1, 2, 3) (3, 1, 2, 4) (3, 1, 2, 5) (3, 1, 2, 6) (3, 1, 3, 1) (3, 1, 3, 2) (3, 1, 3, 3) (3, 1, 3, 4) (3, 1, 3, 5) (3, 1, 3, 6) (3, 1, 4, 1) (3, 1, 4, 2) (3, 1, 4, 3) (3, 1, 4, 4) (3, 1, 4, 5) (3, 1, 4, 6) (3, 1, 5, 1) (3, 1, 5, 2) (3, 1, 5, 3) (3, 1, 5, 4) (3, 1, 5, 5) (3, 1, 5, 6) (3, 1, 6, 1) (3, 1, 6, 2) (3, 1, 6, 3) (3, 1, 6, 4) (3, 1, 6, 5) (3, 1, 6, 6) (3, 2, 1, 1) (3, 2, 1, 2) (3, 2, 1, 3) (3, 2, 1, 4) (3, 2, 1, 5) (3, 2, 1, 6) (3, 2, 2, 1) (3, 2, 2, 2) (3, 2, 2, 3) (3, 2, 2, 4) (3, 2, 2, 5) (3, 2, 2, 6) (3, 2, 3, 1) (3, 2, 3, 2) (3, 2, 3, 3) (3, 2, 3, 4) (3, 2, 3, 5) (3, 2, 3, 6) (3, 2, 4, 1) (3, 2, 4, 2) (3, 2, 4, 3) (3, 2, 4, 4) (3, 2, 4, 5) (3, 2, 4, 6) (3, 2, 5, 1) (3, 2, 5, 2) (3, 2, 5, 3) (3, 2, 5, 4) (3, 2, 5, 5) (3, 2, 5, 6) (3, 2, 6, 1) (3, 2, 6, 2) (3, 2, 6, 3) (3, 2, 6, 4) (3, 2, 6, 5) (3, 2, 6, 6) (3, 3, 1, 1) (3, 3, 1, 2) (3, 3, 1, 3) (3, 3, 1, 4) (3, 3, 1, 5) (3, 3, 1, 6) (3, 3, 2, 1) (3, 3, 2, 2) (3, 3, 2, 3) (3, 3, 2, 4) (3, 3, 2, 5) (3, 3, 2, 6) (3, 3, 3, 1) (3, 3, 3, 2) (3, 3, 3, 3) (3, 3, 3, 4) (3, 3, 3, 5) (3, 3, 3, 6) (3, 3, 4, 1) (3, 3, 4, 2) (3, 3, 4, 3) (3, 3, 4, 4) (3, 3, 4, 5) (3, 3, 4, 6) (3, 3, 5, 1) (3, 3, 5, 2) (3, 3, 5, 3) (3, 3, 5, 4) (3, 3, 5, 5) (3, 3, 5, 6) (3, 3, 6, 1) (3, 3, 6, 2) (3, 3, 6, 3) (3, 3, 6, 4) (3, 3, 6, 5) (3, 3, 6, 6) (3, 4, 1, 1) (3, 4, 1, 2) (3, 4, 1, 3) (3, 4, 1, 4) (3, 4, 1, 5) (3, 4, 1, 6) (3, 4, 2, 1) (3, 4, 2, 2) (3, 4, 2, 3) (3, 4, 2, 4) (3, 4, 2, 5) (3, 4, 2, 6) (3, 4, 3, 1) (3, 4, 3, 2) (3, 4, 3, 3) (3, 4, 3, 4) (3, 4, 3, 5) (3, 4, 3, 6) (3, 4, 4, 1) (3, 4, 4, 2) (3, 4, 4, 3) (3, 4, 4, 4) (3, 4, 4, 5) (3, 4, 4, 6) (3, 4, 5, 1) (3, 4, 5, 2) (3, 4, 5, 3) (3, 4, 5, 4) (3, 4, 5, 5) (3, 4, 5, 6) (3, 4, 6, 1) (3, 4, 6, 2) (3, 4, 6, 3) (3, 4, 6, 4) (3, 4, 6, 5) (3, 4, 6, 6) (3, 5, 1, 1) (3, 5, 1, 2) (3, 5, 1, 3) (3, 5, 1, 4) (3, 5, 1, 5) (3, 5, 1, 6) (3, 5, 2, 1) (3, 5, 2, 2) (3, 5, 2, 3) (3, 5, 2, 4) (3, 5, 2, 5) (3, 5, 2, 6) (3, 5, 3, 1) (3, 5, 3, 2) (3, 5, 3, 3) (3, 5, 3, 4) (3, 5, 3, 5) (3, 5, 3, 6) (3, 5, 4, 1) (3, 5, 4, 2) (3, 5, 4, 3) (3, 5, 4, 4) (3, 5, 4, 5) (3, 5, 4, 6) (3, 5, 5, 1) (3, 5, 5, 2) (3, 5, 5, 3) (3, 5, 5, 4) (3, 5, 5, 5) (3, 5, 5, 6) (3, 5, 6, 1) (3, 5, 6, 2) (3, 5, 6, 3) (3, 5, 6, 4) (3, 5, 6, 5) (3, 5, 6, 6) (3, 6, 1, 1) (3, 6, 1, 2) (3, 6, 1, 3) (3, 6, 1, 4) (3, 6, 1, 5) (3, 6, 1, 6) (3, 6, 2, 1) (3, 6, 2, 2) (3, 6, 2, 3) (3, 6, 2, 4) (3, 6, 2, 5) (3, 6, 2, 6) (3, 6, 3, 1) (3, 6, 3, 2) (3, 6, 3, 3) (3, 6, 3, 4) (3, 6, 3, 5) (3, 6, 3, 6) (3, 6, 4, 1) (3, 6, 4, 2) (3, 6, 4, 3) (3, 6, 4, 4) (3, 6, 4, 5) (3, 6, 4, 6) (3, 6, 5, 1) (3, 6, 5, 2) (3, 6, 5, 3) (3, 6, 5, 4) (3, 6, 5, 5) (3, 6, 5, 6) (3, 6, 6, 1) (3, 6, 6, 2) (3, 6, 6, 3) (3, 6, 6, 4) (3, 6, 6, 5) (3, 6, 6, 6) (4, 1, 1, 1) (4, 1, 1, 2) (4, 1, 1, 3) (4, 1, 1, 4) (4, 1, 1, 5) (4, 1, 1, 6) (4, 1, 2, 1) (4, 1, 2, 2) (4, 1, 2, 3) (4, 1, 2, 4) (4, 1, 2, 5) (4, 1, 2, 6) (4, 1, 3, 1) (4, 1, 3, 2) (4, 1, 3, 3) (4, 1, 3, 4) (4, 1, 3, 5) (4, 1, 3, 6) (4, 1, 4, 1) (4, 1, 4, 2) (4, 1, 4, 3) (4, 1, 4, 4) (4, 1, 4, 5) (4, 1, 4, 6) (4, 1, 5, 1) (4, 1, 5, 2) (4, 1, 5, 3) (4, 1, 5, 4) (4, 1, 5, 5) (4, 1, 5, 6) (4, 1, 6, 1) (4, 1, 6, 2) (4, 1, 6, 3) (4, 1, 6, 4) (4, 1, 6, 5) (4, 1, 6, 6) (4, 2, 1, 1) (4, 2, 1, 2) (4, 2, 1, 3) (4, 2, 1, 4) (4, 2, 1, 5) (4, 2, 1, 6) (4, 2, 2, 1) (4, 2, 2, 2) (4, 2, 2, 3) (4, 2, 2, 4) (4, 2, 2, 5) (4, 2, 2, 6) (4, 2, 3, 1) (4, 2, 3, 2) (4, 2, 3, 3) (4, 2, 3, 4) (4, 2, 3, 5) (4, 2, 3, 6) (4, 2, 4, 1) (4, 2, 4, 2) (4, 2, 4, 3) (4, 2, 4, 4) (4, 2, 4, 5) (4, 2, 4, 6) (4, 2, 5, 1) (4, 2, 5, 2) (4, 2, 5, 3) (4, 2, 5, 4) (4, 2, 5, 5) (4, 2, 5, 6) (4, 2, 6, 1) (4, 2, 6, 2) (4, 2, 6, 3) (4, 2, 6, 4) (4, 2, 6, 5) (4, 2, 6, 6) (4, 3, 1, 1) (4, 3, 1, 2) (4, 3, 1, 3) (4, 3, 1, 4) (4, 3, 1, 5) (4, 3, 1, 6) (4, 3, 2, 1) (4, 3, 2, 2) (4, 3, 2, 3) (4, 3, 2, 4) (4, 3, 2, 5) (4, 3, 2, 6) (4, 3, 3, 1) (4, 3, 3, 2) (4, 3, 3, 3) (4, 3, 3, 4) (4, 3, 3, 5) (4, 3, 3, 6) (4, 3, 4, 1) (4, 3, 4, 2) (4, 3, 4, 3) (4, 3, 4, 4) (4, 3, 4, 5) (4, 3, 4, 6) (4, 3, 5, 1) (4, 3, 5, 2) (4, 3, 5, 3) (4, 3, 5, 4) (4, 3, 5, 5) (4, 3, 5, 6) (4, 3, 6, 1) (4, 3, 6, 2) (4, 3, 6, 3) (4, 3, 6, 4) (4, 3, 6, 5) (4, 3, 6, 6) (4, 4, 1, 1) (4, 4, 1, 2) (4, 4, 1, 3) (4, 4, 1, 4) (4, 4, 1, 5) (4, 4, 1, 6) (4, 4, 2, 1) (4, 4, 2, 2) (4, 4, 2, 3) (4, 4, 2, 4) (4, 4, 2, 5) (4, 4, 2, 6) (4, 4, 3, 1) (4, 4, 3, 2) (4, 4, 3, 3) (4, 4, 3, 4) (4, 4, 3, 5) (4, 4, 3, 6) (4, 4, 4, 1) (4, 4, 4, 2) (4, 4, 4, 3) (4, 4, 4, 4) (4, 4, 4, 5) (4, 4, 4, 6) (4, 4, 5, 1) (4, 4, 5, 2) (4, 4, 5, 3) (4, 4, 5, 4) (4, 4, 5, 5) (4, 4, 5, 6) (4, 4, 6, 1) (4, 4, 6, 2) (4, 4, 6, 3) (4, 4, 6, 4) (4, 4, 6, 5) (4, 4, 6, 6) (4, 5, 1, 1) (4, 5, 1, 2) (4, 5, 1, 3) (4, 5, 1, 4) (4, 5, 1, 5) (4, 5, 1, 6) (4, 5, 2, 1) (4, 5, 2, 2) (4, 5, 2, 3) (4, 5, 2, 4) (4, 5, 2, 5) (4, 5, 2, 6) (4, 5, 3, 1) (4, 5, 3, 2) (4, 5, 3, 3) (4, 5, 3, 4) (4, 5, 3, 5) (4, 5, 3, 6) (4, 5, 4, 1) (4, 5, 4, 2) (4, 5, 4, 3) (4, 5, 4, 4) (4, 5, 4, 5) (4, 5, 4, 6) (4, 5, 5, 1) (4, 5, 5, 2) (4, 5, 5, 3) (4, 5, 5, 4) (4, 5, 5, 5) (4, 5, 5, 6) (4, 5, 6, 1) (4, 5, 6, 2) (4, 5, 6, 3) (4, 5, 6, 4) (4, 5, 6, 5) (4, 5, 6, 6) (4, 6, 1, 1) (4, 6, 1, 2) (4, 6, 1, 3) (4, 6, 1, 4) (4, 6, 1, 5) (4, 6, 1, 6) (4, 6, 2, 1) (4, 6, 2, 2) (4, 6, 2, 3) (4, 6, 2, 4) (4, 6, 2, 5) (4, 6, 2, 6) (4, 6, 3, 1) (4, 6, 3, 2) (4, 6, 3, 3) (4, 6, 3, 4) (4, 6, 3, 5) (4, 6, 3, 6) (4, 6, 4, 1) (4, 6, 4, 2) (4, 6, 4, 3) (4, 6, 4, 4) (4, 6, 4, 5) (4, 6, 4, 6) (4, 6, 5, 1) (4, 6, 5, 2) (4, 6, 5, 3) (4, 6, 5, 4) (4, 6, 5, 5) (4, 6, 5, 6) (4, 6, 6, 1) (4, 6, 6, 2) (4, 6, 6, 3) (4, 6, 6, 4) (4, 6, 6, 5) (4, 6, 6, 6) (5, 1, 1, 1) (5, 1, 1, 2) (5, 1, 1, 3) (5, 1, 1, 4) (5, 1, 1, 5) (5, 1, 1, 6) (5, 1, 2, 1) (5, 1, 2, 2) (5, 1, 2, 3) (5, 1, 2, 4) (5, 1, 2, 5) (5, 1, 2, 6) (5, 1, 3, 1) (5, 1, 3, 2) (5, 1, 3, 3) (5, 1, 3, 4) (5, 1, 3, 5) (5, 1, 3, 6) (5, 1, 4, 1) (5, 1, 4, 2) (5, 1, 4, 3) (5, 1, 4, 4) (5, 1, 4, 5) (5, 1, 4, 6) (5, 1, 5, 1) (5, 1, 5, 2) (5, 1, 5, 3) (5, 1, 5, 4) (5, 1, 5, 5) (5, 1, 5, 6) (5, 1, 6, 1) (5, 1, 6, 2) (5, 1, 6, 3) (5, 1, 6, 4) (5, 1, 6, 5) (5, 1, 6, 6) (5, 2, 1, 1) (5, 2, 1, 2) (5, 2, 1, 3) (5, 2, 1, 4) (5, 2, 1, 5) (5, 2, 1, 6) (5, 2, 2, 1) (5, 2, 2, 2) (5, 2, 2, 3) (5, 2, 2, 4) (5, 2, 2, 5) (5, 2, 2, 6) (5, 2, 3, 1) (5, 2, 3, 2) (5, 2, 3, 3) (5, 2, 3, 4) (5, 2, 3, 5) (5, 2, 3, 6) (5, 2, 4, 1) (5, 2, 4, 2) (5, 2, 4, 3) (5, 2, 4, 4) (5, 2, 4, 5) (5, 2, 4, 6) (5, 2, 5, 1) (5, 2, 5, 2) (5, 2, 5, 3) (5, 2, 5, 4) (5, 2, 5, 5) (5, 2, 5, 6) (5, 2, 6, 1) (5, 2, 6, 2) (5, 2, 6, 3) (5, 2, 6, 4) (5, 2, 6, 5) (5, 2, 6, 6) (5, 3, 1, 1) (5, 3, 1, 2) (5, 3, 1, 3) (5, 3, 1, 4) (5, 3, 1, 5) (5, 3, 1, 6) (5, 3, 2, 1) (5, 3, 2, 2) (5, 3, 2, 3) (5, 3, 2, 4) (5, 3, 2, 5) (5, 3, 2, 6) (5, 3, 3, 1) (5, 3, 3, 2) (5, 3, 3, 3) (5, 3, 3, 4) (5, 3, 3, 5) (5, 3, 3, 6) (5, 3, 4, 1) (5, 3, 4, 2) (5, 3, 4, 3) (5, 3, 4, 4) (5, 3, 4, 5) (5, 3, 4, 6) (5, 3, 5, 1) (5, 3, 5, 2) (5, 3, 5, 3) (5, 3, 5, 4) (5, 3, 5, 5) (5, 3, 5, 6) (5, 3, 6, 1) (5, 3, 6, 2) (5, 3, 6, 3) (5, 3, 6, 4) (5, 3, 6, 5) (5, 3, 6, 6) (5, 4, 1, 1) (5, 4, 1, 2) (5, 4, 1, 3) (5, 4, 1, 4) (5, 4, 1, 5) (5, 4, 1, 6) (5, 4, 2, 1) (5, 4, 2, 2) (5, 4, 2, 3) (5, 4, 2, 4) (5, 4, 2, 5) (5, 4, 2, 6) (5, 4, 3, 1) (5, 4, 3, 2) (5, 4, 3, 3) (5, 4, 3, 4) (5, 4, 3, 5) (5, 4, 3, 6) (5, 4, 4, 1) (5, 4, 4, 2) (5, 4, 4, 3) (5, 4, 4, 4) (5, 4, 4, 5) (5, 4, 4, 6) (5, 4, 5, 1) (5, 4, 5, 2) (5, 4, 5, 3) (5, 4, 5, 4) (5, 4, 5, 5) (5, 4, 5, 6) (5, 4, 6, 1) (5, 4, 6, 2) (5, 4, 6, 3) (5, 4, 6, 4) (5, 4, 6, 5) (5, 4, 6, 6) (5, 5, 1, 1) (5, 5, 1, 2) (5, 5, 1, 3) (5, 5, 1, 4) (5, 5, 1, 5) (5, 5, 1, 6) (5, 5, 2, 1) (5, 5, 2, 2) (5, 5, 2, 3) (5, 5, 2, 4) (5, 5, 2, 5) (5, 5, 2, 6) (5, 5, 3, 1) (5, 5, 3, 2) (5, 5, 3, 3) (5, 5, 3, 4) (5, 5, 3, 5) (5, 5, 3, 6) (5, 5, 4, 1) (5, 5, 4, 2) (5, 5, 4, 3) (5, 5, 4, 4) (5, 5, 4, 5) (5, 5, 4, 6) (5, 5, 5, 1) (5, 5, 5, 2) (5, 5, 5, 3) (5, 5, 5, 4) (5, 5, 5, 5) (5, 5, 5, 6) (5, 5, 6, 1) (5, 5, 6, 2) (5, 5, 6, 3) (5, 5, 6, 4) (5, 5, 6, 5) (5, 5, 6, 6) (5, 6, 1, 1) (5, 6, 1, 2) (5, 6, 1, 3) (5, 6, 1, 4) (5, 6, 1, 5) (5, 6, 1, 6) (5, 6, 2, 1) (5, 6, 2, 2) (5, 6, 2, 3) (5, 6, 2, 4) (5, 6, 2, 5) (5, 6, 2, 6) (5, 6, 3, 1) (5, 6, 3, 2) (5, 6, 3, 3) (5, 6, 3, 4) (5, 6, 3, 5) (5, 6, 3, 6) (5, 6, 4, 1) (5, 6, 4, 2) (5, 6, 4, 3) (5, 6, 4, 4) (5, 6, 4, 5) (5, 6, 4, 6) (5, 6, 5, 1) (5, 6, 5, 2) (5, 6, 5, 3) (5, 6, 5, 4) (5, 6, 5, 5) (5, 6, 5, 6) (5, 6, 6, 1) (5, 6, 6, 2) (5, 6, 6, 3) (5, 6, 6, 4) (5, 6, 6, 5) (5, 6, 6, 6) (6, 1, 1, 1) (6, 1, 1, 2) (6, 1, 1, 3) (6, 1, 1, 4) (6, 1, 1, 5) (6, 1, 1, 6) (6, 1, 2, 1) (6, 1, 2, 2) (6, 1, 2, 3) (6, 1, 2, 4) (6, 1, 2, 5) (6, 1, 2, 6) (6, 1, 3, 1) (6, 1, 3, 2) (6, 1, 3, 3) (6, 1, 3, 4) (6, 1, 3, 5) (6, 1, 3, 6) (6, 1, 4, 1) (6, 1, 4, 2) (6, 1, 4, 3) (6, 1, 4, 4) (6, 1, 4, 5) (6, 1, 4, 6) (6, 1, 5, 1) (6, 1, 5, 2) (6, 1, 5, 3) (6, 1, 5, 4) (6, 1, 5, 5) (6, 1, 5, 6) (6, 1, 6, 1) (6, 1, 6, 2) (6, 1, 6, 3) (6, 1, 6, 4) (6, 1, 6, 5) (6, 1, 6, 6) (6, 2, 1, 1) (6, 2, 1, 2) (6, 2, 1, 3) (6, 2, 1, 4) (6, 2, 1, 5) (6, 2, 1, 6) (6, 2, 2, 1) (6, 2, 2, 2) (6, 2, 2, 3) (6, 2, 2, 4) (6, 2, 2, 5) (6, 2, 2, 6) (6, 2, 3, 1) (6, 2, 3, 2) (6, 2, 3, 3) (6, 2, 3, 4) (6, 2, 3, 5) (6, 2, 3, 6) (6, 2, 4, 1) (6, 2, 4, 2) (6, 2, 4, 3) (6, 2, 4, 4) (6, 2, 4, 5) (6, 2, 4, 6) (6, 2, 5, 1) (6, 2, 5, 2) (6, 2, 5, 3) (6, 2, 5, 4) (6, 2, 5, 5) (6, 2, 5, 6) (6, 2, 6, 1) (6, 2, 6, 2) (6, 2, 6, 3) (6, 2, 6, 4) (6, 2, 6, 5) (6, 2, 6, 6) (6, 3, 1, 1) (6, 3, 1, 2) (6, 3, 1, 3) (6, 3, 1, 4) (6, 3, 1, 5) (6, 3, 1, 6) (6, 3, 2, 1) (6, 3, 2, 2) (6, 3, 2, 3) (6, 3, 2, 4) (6, 3, 2, 5) (6, 3, 2, 6) (6, 3, 3, 1) (6, 3, 3, 2) (6, 3, 3, 3) (6, 3, 3, 4) (6, 3, 3, 5) (6, 3, 3, 6) (6, 3, 4, 1) (6, 3, 4, 2) (6, 3, 4, 3) (6, 3, 4, 4) (6, 3, 4, 5) (6, 3, 4, 6) (6, 3, 5, 1) (6, 3, 5, 2) (6, 3, 5, 3) (6, 3, 5, 4) (6, 3, 5, 5) (6, 3, 5, 6) (6, 3, 6, 1) (6, 3, 6, 2) (6, 3, 6, 3) (6, 3, 6, 4) (6, 3, 6, 5) (6, 3, 6, 6) (6, 4, 1, 1) (6, 4, 1, 2) (6, 4, 1, 3) (6, 4, 1, 4) (6, 4, 1, 5) (6, 4, 1, 6) (6, 4, 2, 1) (6, 4, 2, 2) (6, 4, 2, 3) (6, 4, 2, 4) (6, 4, 2, 5) (6, 4, 2, 6) (6, 4, 3, 1) (6, 4, 3, 2) (6, 4, 3, 3) (6, 4, 3, 4) (6, 4, 3, 5) (6, 4, 3, 6) (6, 4, 4, 1) (6, 4, 4, 2) (6, 4, 4, 3) (6, 4, 4, 4) (6, 4, 4, 5) (6, 4, 4, 6) (6, 4, 5, 1) (6, 4, 5, 2) (6, 4, 5, 3) (6, 4, 5, 4) (6, 4, 5, 5) (6, 4, 5, 6) (6, 4, 6, 1) (6, 4, 6, 2) (6, 4, 6, 3) (6, 4, 6, 4) (6, 4, 6, 5) (6, 4, 6, 6) (6, 5, 1, 1) (6, 5, 1, 2) (6, 5, 1, 3) (6, 5, 1, 4) (6, 5, 1, 5) (6, 5, 1, 6) (6, 5, 2, 1) (6, 5, 2, 2) (6, 5, 2, 3) (6, 5, 2, 4) (6, 5, 2, 5) (6, 5, 2, 6) (6, 5, 3, 1) (6, 5, 3, 2) (6, 5, 3, 3) (6, 5, 3, 4) (6, 5, 3, 5) (6, 5, 3, 6) (6, 5, 4, 1) (6, 5, 4, 2) (6, 5, 4, 3) (6, 5, 4, 4) (6, 5, 4, 5) (6, 5, 4, 6) (6, 5, 5, 1) (6, 5, 5, 2) (6, 5, 5, 3) (6, 5, 5, 4) (6, 5, 5, 5) (6, 5, 5, 6) (6, 5, 6, 1) (6, 5, 6, 2) (6, 5, 6, 3) (6, 5, 6, 4) (6, 5, 6, 5) (6, 5, 6, 6) (6, 6, 1, 1) (6, 6, 1, 2) (6, 6, 1, 3) (6, 6, 1, 4) (6, 6, 1, 5) (6, 6, 1, 6) (6, 6, 2, 1) (6, 6, 2, 2) (6, 6, 2, 3) (6, 6, 2, 4) (6, 6, 2, 5) (6, 6, 2, 6) (6, 6, 3, 1) (6, 6, 3, 2) (6, 6, 3, 3) (6, 6, 3, 4) (6, 6, 3, 5) (6, 6, 3, 6) (6, 6, 4, 1) (6, 6, 4, 2) (6, 6, 4, 3) (6, 6, 4, 4) (6, 6, 4, 5) (6, 6, 4, 6) (6, 6, 5, 1) (6, 6, 5, 2) (6, 6, 5, 3) (6, 6, 5, 4) (6, 6, 5, 5) (6, 6, 5, 6) (6, 6, 6, 1) (6, 6, 6, 2) (6, 6, 6, 3) (6, 6, 6, 4) (6, 6, 6, 5) (6, 6, 6, 6)
Simulation and probability with 24 times 2 dice¶
In [12]:
n = 1000
count = 0
for _ in range(n):
for __ in range(24):
seq = [randint(1,6), randint(1,6)]
if [6,6] == seq:
count += 1
break
print(count/n)
0.488
In [13]:
6**4, 36**24
Out[13]:
(1296, 22452257707354557240087211123792674816)
Checking all possibilities and counting the number of double 6 is impossible, so we need to use math!
In [14]:
1-(35/36)**24
Out[14]:
0.4914038761309034
Exercise: For practice, write a complete program (similar to the one above) for the case of two dice rolled 24 times!
In [15]:
print("The value of __name__ is:", repr(__name__))
The value of __name__ is: '__main__'
Fill in and rewrite the next code:
In [16]:
def simulation():
"""
Describe the function here!
"""
print("This is the simulation function!")
def probability():
"""
Describe the function here!
"""
print("This is the probability function!")
def main():
print("Call here the functions!")
simulation()
probability()
if __name__ == "__main__":
# execute only if run as a script
main()
Call here the functions! This is the simulation function! This is the probability function!
In [ ]: