What is a Monte Carlo simulation?

Monte Carlo Simulations in Python

Izzy Weber

Curriculum Manager, DataCamp

Simulations and Monte Carlo simulations

Simulations:

Experiments that attempt to imitate reality
Often use computer programs

Monte Carlo simulations:

Used to predict the probability of different outcomes impacted by the presence of random variables
Rely on repeated random sampling to obtain numerical results
Results are stochastic since the model relies on random sampling

Simulation example

Rolling six-sided die

Tom rolls a fair six-sided die $n$ times
After each roll, he records the outcome
He then places the die in a bag and selects a new one for the next roll

Questions

How many dice will Tom collect after $n$ rolls?
What will be the mean outcome after $n$ rolls?

a hand rolling a die

Simulating Tom's outcomes

total_dice: the number of dice in Tom's bag after $n$ rolls
mean_point_dice: the mean of all outcomes after $n$ rolls

import random
import numpy as np


def roll_dice(n, seed):
    random.seed(seed)
    total_dice = 0
    point_dice = []

    for i in range(n):
        total_dice += 1
        point_dice.append(random.randint(1, 6))

    mean_point_dice = np.mean(point_dice)

    return([total_dice, mean_point_dice])

Simulation results

Simulation One:

seed=1231

print(roll_dice(10, seed))

print(roll_dice(100, seed))

print(roll_dice(1000, seed))

print(roll_dice(10000, seed))

Simulation Two:

seed=3124
print(roll_dice(10, seed))
print(roll_dice(100, seed))
print(roll_dice(1000, seed))
print(roll_dice(10000, seed))

Results:

[10, 3.6]

[100, 3.5]

[1000, 3.495]

[10000, 3.503]

Results:

[10, 3.8]
[100, 3.28]
[1000, 3.474]
[10000, 3.5508]

The Law of Large Numbers

As the number of identically distributed, randomly generated variables increases, their sample mean approaches their theoretical mean.

Simulation Three (seed = 3124):

print(roll_dice(100000, seed))
print(roll_dice(500000, seed))
print(roll_dice(1000000, seed))

Results:

[100000, 3.51344]
[500000, 3.50428]
[1000000, 3.501995]

Let's practice!

Monte Carlo Simulations in Python