This is the final project of Intro to Probability Part 1

In [1]:

import random
import pandas as pd

Imagine that we will play a role-playing game. First, we need to know how throw a dice.

Try to do alone:

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In [2]:

def dice():
    return random.randint(1,6)
dice()

Out[2]:

Suppose that you attack your enemy. Define a code to attack that depend on the amount of dice.

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In [3]:

def attack(number_of_dice):
    return sum([dice() for _ in range(number_of_dice)])

In [4]:

attack(3)

Out[4]:

Now that we know how to attack, we begin a battle.

We call "enemy_armour" to number that represent the power of the armour of your enemy and "number_of_dice" to the amount of dice when attack.

Define a battle that return true if you win the battle (the sum of the dice>enemy armour) and false in otherwise.

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In [69]:

def battle(number_of_dice,enemy_armour):
    '''Returns True if you beat the enemy and False otherwise'''
    return attack(number_of_dice)>enemy_armour

Suppose that the enemy has an armour with power 8 and you attack with 2 dice. How is the battle?

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In [70]:

def battle_1():
    return battle(2,8)

and if the enemy has an armour with power 12 and you attack with 3 dice?

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In [71]:

def battle_2():
    return battle(3,12)

What previous battle will more likely than you win? We try to calculate the empirical probability.

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In [72]:

def empiric_prob(experiment,n=10000):
    return sum([experiment() for _ in range(n)])/n

In [73]:

empiric_prob(battle_1)

Out[73]:

0.2781

In [74]:

empiric_prob(battle_2)

Out[74]:

0.2584

We will compare the empirical probability with the theoretical:

We begin with the battle of 2 dice and 8 in the enemy armour. Then we need that sum of the dice is more than 8.

In [75]:

amount = 0
for x in range(1,7,1):
    for y in range(1,7,1):
        if x+y>8:
            amount=amount+1
print(amount)

Then we have $\frac{10}{36}=\frac{5}{18}$, that it this is:

In [76]:

5/18

Out[76]:

0.2777777777777778

In [77]:

abs(empiric_prob(battle_1)-5/18)

Out[77]:

0.002922222222222215

Is close to "empiric_prob(battle_2)", but it can be better. What will we can do to better the empirical probability?

In [78]:

emp_prob = empiric_prob(battle_1,n=100000)
emp_prob

Out[78]:

0.27883

In [79]:

abs(emp_prob- 5/18)

Out[79]:

0.0010522222222222322

We can do a table to compare the result with different n:

In [80]:

times = [10000, 100000, 1000000, 10000000]
experiments = []
for t in times:
    experiments.append(empiric_prob(battle_1,t))
dif = []    
for i in range(len(times)): 
    dif.append(abs(experiments[i]-5/18))   
data = {'Times':['10000', '100000', '1000000', '10000000'], 'Empirical Probability': experiments,
        'Difference': dif}
df = pd.DataFrame(data)
df

Out[80]:

	Times	Empirical Probability	Difference
0	10000	0.270700	0.007078
1	100000	0.276150	0.001628
2	1000000	0.277570	0.000208
3	10000000	0.277962	0.000184

Try to do the compare in the battle 2

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In [83]:

amount = 0
for x in range(1,7,1):
    for y in range(1,7,1):
        for z in range(1,7,1):
            if x+y+z>12:
                amount=amount+1
times = [10000, 100000, 1000000, 10000000]
experiments = []
for t in times:
    experiments.append(empiric_prob(battle_2,t))
dif = []    
for i in range(len(times)): 
    dif.append(abs(experiments[i]-amount/6**3))   
data = {'Times':['10000', '100000', '1000000', '10000000'], 'Empirical Probability': experiments,
        'Difference': dif}
df = pd.DataFrame(data)
df

Out[83]:

	Times	Empirical Probability	Difference
0	10000	0.258000	0.001259
1	100000	0.256750	0.002509
2	1000000	0.259447	0.000188
3	10000000	0.259383	0.000124

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MguadapvMguadapv

Dice Exercises