1. Convergence
The average results of repeated random trials will converge to the expected value over a large number of trials.
Law of Large Numbers Simulation
Simulations
Coin Flip Simulation
Parameters
Current Probability:0.00%
Expected Probability:50.00%
Results
Understanding the Results
- The blue line shows the probability of getting heads as trials increase
- The red dashed line shows the theoretical probability (50%)
- Notice how the experimental probability converges to 50% over time
- This demonstrates the Law of Large Numbers in action
Dice Roll Simulation
Parameters
Current Average:0.00
Expected Average:3.50
Results
Understanding the Results
- The blue line shows the average dice roll value over time
- The red dashed line shows the expected average (3.5)
- Notice how the experimental average converges to 3.5 as trials increase
- This demonstrates how sample means converge to the true population mean
Card Draw Simulation
Parameters
Current Probability:0.00%
Expected Probability:7.69%
Results
Understanding the Results
- The blue line shows the probability of drawing an ace over time
- The red dashed line shows the theoretical probability (4/52 ≈ 7.69%)
- Notice how the experimental probability converges to 7.69% as trials increase
- This demonstrates the Law of Large Numbers for a less intuitive probability
Learn More
Understanding the Law of Large Numbers
Overview
The Law of Large Numbers is a fundamental principle in probability theory and statistics that describes how the average of results from repeated experiments will converge to the expected value as the number of trials increases.
Mathematical Foundation
Let be a sequence of independent and identically distributed random variables with expected value . The sample average is:
The law states that as approaches infinity:
for any
Key Concepts
2. Independence
The trials must be independent of each other - the outcome of one trial does not affect the outcomes of other trials.
3. Sample Size
Larger sample sizes lead to more stable and accurate estimates of the true probability or expected value.
4. Variability
While individual trials may vary significantly, the average becomes more stable as the number of trials increases.
Practical Applications
- Insurance: Calculating premiums based on historical claim data
- Quality Control: Monitoring manufacturing processes
- Gambling: Understanding expected returns in games of chance
- Scientific Research: Validating experimental results
- Polling: Determining appropriate sample sizes for surveys
Simulation Insights
Our three simulations demonstrate different aspects of the law:
- Coin Flip: Shows convergence to a simple probability (50%)
- Dice Roll: Demonstrates convergence to an expected value (3.5)
- Card Draw: Illustrates convergence for a more complex probability