import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# this generates a dataframe showing all the possible arragements of items behind the door
possibility_1 = ['goat', 'goat', 'prize']
possibility_2 = ['goat', 'prize', 'goat']
possibility_3 = ['prize', 'goat', 'goat']
df = pd.DataFrame([possibility_1, possibility_2, possibility_3], columns = ['door_1', 'door_2', 'door_3'])
df.index = ['possibility 1:','possibility 2:', 'possibility 3:']
print('\n\nHere are all the possible arrangements of the items behind the doors:')
display(df)
# this generates an output that shows all possible outcomes for all possible choices
doors = np.arange(1,4)
for door in doors:
door_choice = door
print('\n\n####################################################################################################################\n')
print('If you chose door number', door, ':\n')
if door_choice == 1:
door_choice = 'door_1'
elif door_choice == 2:
door_choice = 'door_2'
elif door_choice == 3:
door_choice = 'door_3'
for i in np.arange(len(df)):
if df[door_choice].iloc[i] == 'goat':
print('\nIf you chose '+str(door_choice)+', and possibility '+str(i+1)+' was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.')
if df[door_choice].iloc[i] == 'prize':
print('\nIf you chose '+str(door_choice)+', and possibility '+str(i+1)+' was true, you would have LOST if you had SWITCHED, and WON if you had STAYED.')
Here are all the possible arrangements of the items behind the doors:
####################################################################################################################
If you chose door number 1 :
If you chose door_1, and possibility 1 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
If you chose door_1, and possibility 2 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
If you chose door_1, and possibility 3 was true, you would have LOST if you had SWITCHED, and WON if you had STAYED.
####################################################################################################################
If you chose door number 2 :
If you chose door_2, and possibility 1 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
If you chose door_2, and possibility 2 was true, you would have LOST if you had SWITCHED, and WON if you had STAYED.
If you chose door_2, and possibility 3 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
####################################################################################################################
If you chose door number 3 :
If you chose door_3, and possibility 1 was true, you would have LOST if you had SWITCHED, and WON if you had STAYED.
If you chose door_3, and possibility 2 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
If you chose door_3, and possibility 3 was true, you would have WON if you had SWITCHED, and LOST if you had STAYED.
display(df)
# setting the number of iterations for the simulation
n_iters = 10000
# creating arrays to store the outcomes of three different strategies: always switching, never switching and randomly switching
always_switch_outcomes = np.array([])
never_switch_outcomes = np.array([])
random_switch_outcomes = np.array([])
# running thousands of trials to see which strategy is most effective
for i in np.arange(n_iters):
df = pd.DataFrame([possibility_1, possibility_2, possibility_3], columns = ['door_1', 'door_2', 'door_3'])
df.index = ['possibility 1:','possibility 2:', 'possibility 3:']
# randomly deciding the arrangement of items behind the doors
state_of_the_world = df.loc[np.random.choice(df.index)]
# picking a first door at random
first_choice = np.random.choice(state_of_the_world.index)
# opening the remaining door which has a goat behind it
remaining = state_of_the_world.index[state_of_the_world.index != first_choice]
remaining = np.random.permutation(remaining)
for r in remaining:
if state_of_the_world.loc[r] == 'goat':
state_of_the_world.loc[r] = 'goat (open)'
break
# choosing to switch or stay based on the three strategies
not_open = state_of_the_world[state_of_the_world != 'goat (open)']
switcher_outcome = not_open.loc[not_open.index[not_open.index !=first_choice][0]]
stayer_outcome = state_of_the_world.loc[first_choice]
random_outcome = np.random.choice(state_of_the_world[state_of_the_world != 'goat (open)'])
# recording the outcome for each strategy
always_switch_outcomes = np.append(always_switch_outcomes,switcher_outcome)
never_switch_outcomes = np.append(never_switch_outcomes,stayer_outcome)
random_switch_outcomes = np.append(random_switch_outcomes,random_outcome)
# calculating the proportion of wins for each strategy
prop_wins_always_switch = np.count_nonzero(always_switch_outcomes == 'prize')/len(always_switch_outcomes)
prop_wins_never_switch = np.count_nonzero(never_switch_outcomes == 'prize')/len(never_switch_outcomes)
prop_wins_random_switch = np.count_nonzero(random_switch_outcomes == 'prize')/len(random_switch_outcomes)
# plotting the outcomes of the simulation
plt.figure(figsize = (10,4))
plt.title('% of wins for 3 different strategies over '+str(n_iters)+' Monty Hall trials')
plt.bar(['Always Switch', 'Never Switch', 'Randomly Switch',],
[prop_wins_always_switch*100, prop_wins_never_switch*100, prop_wins_random_switch*100],
color = ['red', 'black', 'cyan'])
plt.xlabel('Strategy')
plt.ylabel('% of wins')
plt.show()
print('There were '+str(n_iters)+' trials.\n')
print('The "Always Switch" strategy won the prize on', prop_wins_always_switch*100,'% of the trials.\n')
print('The "Never Switch" strategy won the prize on', prop_wins_never_switch*100,'% of the trials.\n')
print('The "Randomly Switch" strategy won the prize on', prop_wins_random_switch*100,'% of the trials.\n')
There were 10000 trials.
The "Always Switch" strategy won the prize on 66.57 % of the trials.
The "Never Switch" strategy won the prize on 33.43 % of the trials.
The "Randomly Switch" strategy won the prize on 50.28 % of the trials.