Random processes and statistical experiments
Random Processes and Experiments
Random Processes and Experiments is a topic in Statistics & Probability in the Common Core State Standards. It is aligned to Standard S-IC.2, which requires students to decide if a specified model is consistent with results from a data-generating process, e.g. using simulation.
Random processes can be simulated to estimate probability; the law of large numbers says long-run proportions approach the true probability.
Theory
- Estimate a probability by the long-run proportion.
- The law of large numbers: proportions approach the true probability as trials grow.
- Random assignment makes groups comparable in an experiment.
Estimate a probability:
How to simulate
- Model each trial with a random device.
- Run many trials.
- Record the proportion of successes.
- Use it to estimate the probability.
Flip many times and take the proportion of heads.
| \(\hat p\) | \(=\) | \(\dfrac{\text{heads}}{\text{flips}}\) |
It approaches the true probability.
| \(\text{proportion}\) | \(\to\) | \(P(\text{heads})=0.5\) |
Random assignment balances other factors, so differences can be attributed to the treatment.
Use the observed proportion.
| \(P(\text{red})\) | \(\approx\) | \(\dfrac{18}{60}=0.3\) |
Common pitfalls
Frequently asked questions
How do you estimate a probability by simulation?
Run many trials and take the proportion of successes.
What is the law of large numbers?
Long-run proportions approach the true probability.
What is random assignment?
Randomly placing subjects into groups so the groups are comparable.
Do short runs match the true probability?
Not necessarily — only long runs stabilize.