In [1]:

```
from random import randint
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
```

This function will generate random rounds for our game. Each round consists of 3 doors. Only one of the doors is correct, other two are wrong.

In [2]:

```
def generate_game(n: int):
game = []
for _ in range(n):
doors = [False] * 3
winner = randint(0, 2) # Choose winning door by random
doors[winner] = True
game.append(doors)
return game
```

This is a helper function that takes a list of 3 doors, looks at the second and third door, and then opens the one with goat (i.e. wrong door). This simulates a host with knowledge of what is behind the doors.

In [3]:

```
def reveal_goat(doors):
# Get from doors 2 and 3 the one which contains goat.
for i in range(1, 3):
if doors[i] == False:
return i
```

**Simulate random choice**

Simulate situation where the player randomly chooses whether to keep his initial choice or switch his choice.

In [4]:

```
def simulate_random_choice(game: list):
wins = 0
attempts = 0
history = []
for doors in game:
attempts += 1
# Host reveals a door with goat.
goat = reveal_goat(doors)
# Player randomly chooses whether to keep initial choice or switch.
new_choice = randint(0, 1)
final_choice = 0 if new_choice == 0 else 2 if goat == 1 else 1
if (doors[final_choice] == True):
wins += 1
history.append(wins / attempts)
return wins, history
```

**Simulate initial choice**

Simulate situation where the player *only* keeps his initial choice and never switches.

In [5]:

```
def simulate_keep_choice(game: list):
wins = 0
attempts = 0
history = []
for doors in game:
attempts += 1
# User does not switch game.
if (doors[0] == True):
wins += 1
history.append(wins / attempts)
return wins, history
```

**Simulate switch choice**

Simulate situation where the player switches his choice everytime.

In [6]:

```
def simulate_switch_choice(game: list):
wins = 0
attempts = 0
history = []
for doors in game:
attempts += 1
# Host reveals a door with goat.
goat = reveal_goat(doors)
# Player switches his doors (here he chooses the non-opened doors).
new_choice = 1 if goat == 2 else 2
if (doors[new_choice] == True):
wins += 1
history.append(wins / attempts)
return wins, history
```

Now the computing begins. This generates $n$ random games for simulation.

In [7]:

```
game = generate_game(1000)
```

Run the three simulations defined above for the generated game.

In [8]:

```
wins_random, history_random = simulate_random_choice(game)
wins_keep, history_keep = simulate_keep_choice(game)
wins_switch, history_switch = simulate_switch_choice(game)
```

And finally, create some fancy graph so we can actually see the result.

In [9]:

```
plt.figure(figsize=(12,8))
plt.plot(history_random, 'r', label="Random switch")
plt.plot(history_keep, 'g', label="Keep initial")
plt.plot(history_switch, 'b', label="Only switch")
plt.legend(loc='upper right')
plt.ylim(0, 1.0)
plt.xlim(0, 1000)
plt.ylabel("Chance", fontsize=16)
plt.gca().yaxis.set_major_formatter(ticker.PercentFormatter(xmax=1.0))
plt.xlabel("Iterations", fontsize=16)
plt.grid(True)
plt.show()
```