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Thanos and Avengers playing roulette

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Words count in article: 1,486 | Reading time ≈ 9

Today I found an interesting revision of Avengers: Endgame. It says:

初代复仇者六人组，每人带一宝石手套，去找灭霸决战，到了草屋，七人围圈而坐，六对一，互相打响指意图消灭对方。灭霸人少，给予先打响指的权利，灭霸打后，超级英雄全化灰就灭霸胜，没全化灰就由一个活着的超级英雄打一个响指，看二分之一消失的概率能不能把灭霸打成灰。没成灰就继续第二回合，灭霸继续优先打响指。问灭霸死的概率是多少?

The original six avengers, each carrying a stone and fight with Thanos. Thanos snaps his fingers first and each avenger has a 50% probability to die. If all of the avengers die, thanos wins. If not, one of the alive avengers snap fingers and thanos has 50% percent to die. The fights will continue until one side wins. the question is: what is the probability of thanos death?

I still didn’t have a mathematical depict of the probability. But I simulate the data to find out the probability of Thanos’ death is about 80%.

Like Dr. Strange, I also find the result from 14000605 times test.

Refer to this repo

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import numpy as np
%pylab inline
from tqdm import tqdm_notebook as tqdm
import seaborn as sns
import pandas as pd
np.random.seed(1111)
#setup figure template

figure_template_path = 'bin'
if figure_template_path not in sys.path:
    sys.path.append(figure_template_path)
from importlib import reload
import figure_template
#force reload of the module
reload(figure_template)
from figure_template import display_dataframe, embed_pdf_figure, embed_pdf_pages,std_plot
Populating the interactive namespace from numpy and matplotlib

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np.random.randint(0,2,size=6)
array([0, 1, 1, 1, 0, 0])

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def check_survive(input_num):
    '''
    input_num: how many people to be decided to survive by stones
    status: 0 means dead, 1 means alive
    '''
    random_output = np.random.randint(0,2,size=input_num)
    alive = random_output.sum()
    dead = random_output.shape[0] - alive
    return alive, dead
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avengers_alive_num = 6
round_num = 0
while True:
    round_num += 1
    avengers_alive_num,avengers_dead_num = check_survive(avengers_alive_num)
    print ('Round {}, avengers alive: {}, dead: {}'.format(round_num,avengers_alive_num,avengers_dead_num) )
    if avengers_alive_num ==0:
        print ('Avengers die first')
        break
    elif np.random.randint(2) ==0:
        print ('Thanos die first')
        break
Round 1, avengers alive: 1, dead: 5
Thanos die first

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def summarize(avengers_alive_num = 6):
    round_num = 0
    while True:
        round_num += 1
        avengers_alive_num,avengers_dead_num = check_survive(avengers_alive_num)
        #print ('Round {}, avengers alive: {}, dead: {}'.format(round_num,avengers_alive_num,avengers_dead_num) )
        if avengers_alive_num ==0:
            #print ('Avengers die first')
            return round_num, 1
        elif np.random.randint(2) ==0:
            #print ('Thanos die first')
            return round_num, 0
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test_time = 14000605
summarize_data = np.ndarray([test_time,2])
for i in tqdm(range(test_time)):
    summarize_data[i] = summarize()
HBox(children=(IntProgress(value=0, max=14000605), HTML(value='')))

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np.savetxt('summarize_data_num_6.txt',summarize_data.astype('uint8'),fmt='%1d')
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fig,ax=plt.subplots(figsize=(8,4))
sns.distplot(summarize_data[:,1],kde=0,ax=ax)
embed_pdf_figure()
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fig,ax=plt.subplots(figsize=(8,4))
sns.distplot(summarize_data[np.where(summarize_data[:,1]==0)][:,0],kde=0,ax=ax,color='r')
sns.distplot(summarize_data[np.where(summarize_data[:,1]==1)][:,0],kde=0,ax=ax,color='m')
embed_pdf_figure()
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print ('Avengers win {}, Thanos win {}'.format(1 - summarize_data[:,1].sum()/test_time,summarize_data[:,1].sum()/test_time) )
Avengers win 0.7940833985388489, Thanos win 0.20591660146115115

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unique_sum = np.unique(summarize_data[np.where(summarize_data[:,1]==0)][:,0],return_counts=1)
df = pd.DataFrame(np.concatenate((unique_sum[1].reshape(-1,1),unique_sum[0].astype('int').reshape(-1,1),unique_sum[1].reshape(-1,1)/test_time),axis=1))
df.iloc[:,1] = np.array(df.iloc[:,1]).astype('int')
df = df.set_index(1)
df.columns = ['win time','win probability']
display_dataframe(df,filename='Avengers win times in {} times'.format(test_time))

Avengers win times in 14000605 times
win time win probability
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1 6.89032e+06 0.492144
2 2.8789e+06 0.205627
3 964446 0.068886
4 281804 0.020128
5 75835 0.00541655
6 19639 0.00140273
7 4988 0.00035627
8 1274 9.09961e-05
9 331 2.36418e-05
10 91 6.49972e-06
11 17 1.21423e-06
12 5 3.57127e-07
13 1 7.14255e-08 
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unique_sum = np.unique(summarize_data[np.where(summarize_data[:,1]==1)][:,0],return_counts=1)
df = pd.DataFrame(np.concatenate((unique_sum[1].reshape(-1,1),unique_sum[0].astype('int').reshape(-1,1),unique_sum[1].reshape(-1,1)/test_time),axis=1))
df.iloc[:,1] = np.array(df.iloc[:,1]).astype('int')
df = df.set_index(1)
df.columns = ['win time','win probability']
display_dataframe(df,filename='Thanos win times in {} times'.format(test_time))

Thanos win times in 14000605 times
win time win probability
1
1 218176 0.0155833
2 1.13563e+06 0.0811129
3 948005 0.0677117
4 402860 0.0287745
5 128810 0.00920032
6 36483 0.00260582
7 9634 0.000688113
8 2508 0.000179135
9 660 4.71408e-05
10 147 1.04995e-05
11 27 1.92849e-06
12 11 7.8568e-07
13 5 3.57127e-07
14 1 7.14255e-08

It’s not fair! In fact, if there are only two avengers, the game will be more fair.

灭霸死的有点惨，其实复仇者初始只有两个人的时候概率稍微对等

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def summarize(avengers_alive_num = 6):
    round_num = 0
    while True:
        round_num += 1
        avengers_alive_num,avengers_dead_num = check_survive(avengers_alive_num)
        #print ('Round {}, avengers alive: {}, dead: {}'.format(round_num,avengers_alive_num,avengers_dead_num) )
        if avengers_alive_num ==0:
            #print ('Avengers die first')
            return round_num, 1
        elif np.random.randint(2) ==0:
            #print ('Thanos die first')
            return round_num, 0
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test_time = 14000605
summarize_data = np.ndarray([test_time,2])
for i in tqdm(range(test_time)):
    summarize_data[i] = summarize(avengers_alive_num =2)
HBox(children=(IntProgress(value=0, max=14000605), HTML(value='')))

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fig,ax=plt.subplots(2,1,figsize=(8,8))
sns.distplot(summarize_data[:,1],kde=0,ax=ax[0])
sns.distplot(summarize_data[np.where(summarize_data[:,1]==0)][:,0],kde=0,ax=ax[1],color='r')
sns.distplot(summarize_data[np.where(summarize_data[:,1]==1)][:,0],kde=0,ax=ax[1],color='m')
embed_pdf_figure()
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print ('Avengers win {}, Thanos win {}'.format(1 - summarize_data[:,1].sum()/test_time,summarize_data[:,1].sum()/test_time) )
Avengers win 0.523922359069483, Thanos win 0.4760776409305169

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unique_sum = np.unique(summarize_data[np.where(summarize_data[:,1]==0)][:,0],return_counts=1)
df = pd.DataFrame(np.concatenate((unique_sum[1].reshape(-1,1),unique_sum[0].astype('int').reshape(-1,1),unique_sum[1].reshape(-1,1)/test_time),axis=1))
df.iloc[:,1] = np.array(df.iloc[:,1]).astype('int')
df = df.set_index(1)
df.columns = ['win time','win probability']
display_dataframe(df,filename='Avengers win times in {} times'.format(test_time))
    
unique_sum = np.unique(summarize_data[np.where(summarize_data[:,1]==1)][:,0],return_counts=1)
df = pd.DataFrame(np.concatenate((unique_sum[1].reshape(-1,1),unique_sum[0].astype('int').reshape(-1,1),unique_sum[1].reshape(-1,1)/test_time),axis=1))
df.iloc[:,1] = np.array(df.iloc[:,1]).astype('int')
df = df.set_index(1)
df.columns = ['win time','win probability']
display_dataframe(df,filename='Thanos win times in {} times'.format(test_time))

Avengers win times in 14000605 times
win time win probability
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1 5.25233e+06 0.37515
2 1.53058e+06 0.109322
3 409989 0.0292837
4 106393 0.00759917
5 26925 0.00192313
6 6764 0.000483122
7 1723 0.000123066
8 377 2.69274e-05
9 102 7.2854e-06
10 35 2.49989e-06
11 11 7.8568e-07
13 1 7.14255e-08

Thanos win times in 14000605 times
win time win probability
1
1 3.49891e+06 0.249911
2 2.18546e+06 0.156097
3 712026 0.0508568
4 199091 0.0142202
5 52039 0.00371691
6 13297 0.000949745
7 3438 0.000245561
8 816 5.82832e-05
9 215 1.53565e-05
10 65 4.64266e-06
11 17 1.21423e-06
12 5 3.57127e-07
13 1 7.14255e-08 
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print ('Avengers win {}, Thanos win {}'.format(1 - summarize_data[:,1].sum()/test_time,summarize_data[:,1].sum()/test_time) )
Avengers win 0.523922359069483, Thanos win 0.4760776409305169

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np.savetxt('summarize_data_num_2.txt',summarize_data.astype('uint8'),fmt='%1d')
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