Monte Carlo Method -random numbers

By | February 13, 2026
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The Monte Carlo method is a technique that uses random numbers to solve complex problems.

For example, consider the following types of problems:

  • Estimating the value of π (pi)
  • Calculating the volume of a complex shape
  • Predicting the price of financial products
  • Performing physical simulations

These problems can be difficult to solve analytically. The Monte Carlo method involves performing experiments using a large number of randomly generated numbers and making statistical estimations based on the results.

Origin of the name: It’s named after Monaco, where gambling games like roulette are played in casinos, evoking the image of using random numbers.

Basic Idea of the Monte Carlo Method

  1. Generate Random Numbers: Generate a large number of random numbers necessary to solve the problem.
  2. Perform Experiments: Run experiments that simulate the problem using the generated random numbers.
  3. Statistical Estimation: Collect the results of the experiments and calculate statistics such as averages or probabilities to estimate the solution to the problem.

Example: Estimating π (pi) (Sample Code)

As a simple example of the Monte Carlo method, let’s look at how to estimate the value of π.

import random

def estimate_pi(num_samples):
  """
  Estimates pi using the Monte Carlo method.

  Args:
    num_samples: The number of random numbers to generate.

  Returns:
    An estimation of pi.
  """
  inside_circle = 0
  for _ in range(num_samples):
    x = random.uniform(-1, 1) # Generate a random number between -1 and 1
    y = random.uniform(-1, 1) # Generate a random number between -1 and 1

    # If the distance from the origin is less than or equal to 1, consider it inside the circle
    if x**2 + y**2 <= 1:
      inside_circle += 1

  # Since the area of the circle / area of the square = π/4, estimate pi.
  pi_estimate = 4 * inside_circle / num_samples
  return pi_estimate

# Estimate pi by specifying the number of random numbers
num_samples = 1000000
pi_value = estimate_pi(num_samples)
print(f"Estimated value of pi: {pi_value}")

Code Explanation:

  1. random.uniform(-1, 1) generates random numbers x and y between -1 and 1.
  2. These x and y can be considered as coordinates of points randomly falling within a square with side length 2.
  3. Points whose distance from the origin (0, 0) is less than or equal to 1 are inside a circle with radius 1.
  4. Generate a large number of random numbers and count the number of points inside_circle that fall within the circle.
  5. The area of the circle is π * r^2 = π. The area of the square is 2 * 2 = 4.
  6. Therefore, the ratio of points inside the circle to the total number of points is equal to π/4.
  7. Consequently, pi_estimate = 4 * inside_circle / num_samples estimates pi.

Key Points:

  • The more random numbers num_samples, the closer the estimation will be to the true value.
  • The Monte Carlo method can take a significant amount of computation time.

Advantages and Disadvantages of the Monte Carlo Method

Advantages:

  • Can solve complex problems
  • Applicable to problems where analytical solutions are difficult
  • Suitable for parallel processing

Disadvantages:

  • May require considerable computation time
  • Estimations contain errors
  • Dependent on the quality of random numbers

The Monte Carlo method is a powerful tool for solving complex problems using random numbers. It’s used in various fields and is expected to be increasingly utilized in the future.

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