Chapter 4_ Monte Carlo in Applications

1940s, Manhattan Project

Monte Carlo Simulation

• Developed in the _____ during the _____

Monte Carlo Casino, Monaco

Monte Carlo Simulation

• Named after the _____ in _____

Monte Carlo Simulation

• Uses random sampling,

• performs many repeated trials,

• observes possible outcomes,

• estimates probabilities or numerical results.

• does not always replace mathematical solutions.

• It is especially useful when uncertainty and complexity make direct calculation difficult.

• With the help of simulation, it gives approximate answers quickly.

Stochastic Simulation

• Monte Carlo simulation is primarily a _____ because it includes

• randomness,

• probability, and

• uncertain outcomes.

• The same Monte Carlo model may produce slightly different results in different runs.

Random Sampling

• Selecting or generating values randomly.

• In Monte Carlo simulation:

  • Random values are generated.

  • The values are used as inputs.

  • The model is evaluated.

  • The process is repeated many times.

Random Number

• Computers use random or pseudorandom numbers to simulate uncertainty.

• helps represent an uncertain event.

random sampling, statistics,

simulate random trials,

random samples

• The combination of _____ and _____;

• instead of solving equations, you _____ and look at the results.

• It is about using _____ to estimate something that is difficult to calculate easily.

Estimate = number of favorable outcomes / total trials

Random Numbers in Monte Carlo formula

Step 1: Define the Problem/Model (Late-to-Work Scenario)

• We want to estimate the probability of arriving late to work (after 9:00 AM) using Monte Carlo simulation.

• The person leaves home at 8:30 AM.

• If total travel time exceeds 30 minutes, they are late.

Step 2: Specify Input Variables (Late-to-Work Scenario)

Uncertain variables are modeled with probabilities:

1. Traffic condition (minutes delay):

• Light traffic: 15 min (40% chance)

• Moderate traffic: 20 min (35% chance)

• Heavy traffic: 30 min (25% chance)

2. Weather condition (additional delay):

• Clear: +0 min (70% chance)

• Rainy: +5 min (30% chance)

3. Traffic lights (random delay per trip):

• Between 0–5 minutes (uniform distribution).

Step 3: Generate Random Inputs/Samples (Late-to-Work Scenario)

We simulate many trips (say 10,000) by randomly generating values for:

• Traffic type

• Weather condition

• Traffic light delay

Trial	Traffic	Weather	Light Delay
1	Light	Clear	2 min
2	Heavy	Rainy	4 min
3	Moderate	Clear	1 min

Step 4: Perform the Trial (Late-to-Work Scenario)

• Use the random values to perform one trial.

• Then repeat the process many times.

• For each trial:

• Total Travel Time =Traffic Time +Weather Delay + Traffic Lights

• If Total Travel Time > 30 minutes, mark as Late.

• Trial 1: Travel Time = 25 minutes → On Time

• Trial 2: Travel Time = 38 minutes → Late

• Trial 3: Travel Time = 28 minutes → On Time

• More trials provide more information about possible outcomes.

Step 5: Collect/Aggregate Results (Late-to-Work Scenario)

After simulating 10,000 trips:

• Suppose 3,600 trials resulted in being late.

• Total trials = 10,000.

Step 6: Analyze the Output (Late-to-Work Scenario)

• Estimated probability of being late:

• 𝑷("Late" )="Number of Late Trials" /"Total Trials" =3600/10000=𝟎.𝟑𝟔

• Estimated probability of being late: 36%

Step 7: Interpretation and Decision Making (Late-to-Work Scenario)

• The worker has a 36% chance of being late.

• If this risk is too high, they could:

  • Leave earlier (e.g., 8:15 AM instead of 8:30).

  • Choose alternate routes.

  • Consider work-from-home during heavy rain days.

Monte Carlo with a Coin Toss

• Suppose we simulate 100 coin tosses.

• Possible outcomes:

• Heads

• Tails

• Estimated probability of heads: 52 / 100 = 0.52 or 52%

Why Repeat the Simulation?

More trials produce better estimates, since one simulation trial is not enough because of randomness.

Application 1: Finance — Investment Risk

• An investment may have uncertain returns.

• Monte Carlo simulation can generate many possible market conditions to estimate:

  • Possible profit

  • Possible loss

  • Investment risk

Application 2: Healthcare — Hospital Planning

• Patient arrivals may be unpredictable.

• Monte Carlo simulation can help estimate:

  • Number of patients

  • Waiting times

  • Required staff

  • Possible resource shortages

Application 3: Disaster Risk — Typhoon or Flood Risk

• Monte Carlo methods can simulate many possible conditions involving:

  • Rainfall

  • Wind speed

  • Flood level

  • Population exposure

• The results can help estimate possible risks.

Application 4: Project Management — Will the Project Finish on Time?

• Activities may experience:

  • Short delays

  • Moderate delays

  • Major delays

• Monte Carlo simulation can generate many possible project scenarios and estimate the probability of:

  • Finishing early

  • Finishing on time

  • Finishing late

Monty Hall Problem

• You choose between three doors: one has a car, the other two have goats.

• The host, Monty Hall, opens one of the other doors, revealing a goat.

• Monty asks whether you want to switch your choice to the remaining door.

• You should always switch — switching gives a 66.67% chance of winning, while not switching gives only 33.3%.

Law of Large Numbers

• A fundamental theorem of probability stating that as the number of trials increases, the sample mean approaches the true expected value.

• Monte Carlo relies heavily on the _____;

• the accuracy of Monte Carlo improves with more random samples.

• With few trials, the estimate is rough; with many trials, the estimate stabilizes near π.

Weak LLN

Convergence in probability.

Strong LLN

Convergence almost surely.

Central Limit Theorem

• If we take repeated samples from any population with finite mean μ (mu) and variance σ² (sigma squared), then the distribution of the sample means approaches a Normal Distribution as sample size increases.

• Works regardless of the population’s original distribution.

• Larger sample sizes → faster convergence. Let’s us calculate confidence intervals or margin of error