Overview
This lecture explains the law of large numbers and emphasizes how large sample sizes lead empirical probabilities to closely match true theoretical probabilities.
Law of Large Numbers
- The law of large numbers states that repeating an experiment many times causes the empirical probability to approach the true probability.
- A sample size of at least 100 trials is recommended to ensure accuracy.
- Large sample sizes make survey or experiment results reflect the actual population probability.
Empirical vs. Theoretical Probability
- Empirical probability is the observed proportion of outcomes in experiments or surveys.
- Theoretical probability is the expected probability based on known possible outcomes.
Importance of Large Samples
- A large sample produces empirical probabilities very close to the theoretical probabilities.
- For example, surveying 50,000 people about a fact will give a proportion close to the population's true knowledge.
- Rolling a die a million times will yield a result for rolling a five close to the theoretical 1/6 probability.
Common Mistakes & Correct Approach
- Using too small a sample, like flipping a coin only ten times, may yield misleading probabilities.
- Proper conclusions about probability require a sufficient number of trials to ensure reliability.
- To improve accuracy, repeat the experiment with many more trials (at least a hundred or more).
Key Terms & Definitions
- Law of Large Numbers — The principle that as the number of trials increases, the empirical probability approaches the theoretical probability.
- Empirical Probability — The probability calculated from experimental data or actual observations.
- Theoretical Probability — The probability computed based on known possible outcomes, assuming randomness and fairness.
Action Items / Next Steps
- Review Example 3 and assess what went wrong in the experiment.
- Practice explaining to others why using a large number of trials gives more accurate probabilities.
- Complete any assigned readings on the law of large numbers before the next lesson.