Algebra 2
Statistics and probability
Random processes and statistical experiments
20 practice questions
0 video lessons
Theory + worked examples
Theory
Random processes produce outcomes governed by chance, which we can simulate:
- 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.
More trials give a proportion closer to the true probability.
The simulated proportion approaches \(50\%\).
Random processes and simulation.
Estimate a probability:
\[P(\text{event})\approx\dfrac{\text{successes}}{\text{trials}}\]
The estimate stabilizes as the number of trials grows.
How to simulate
- Model each trial with a random device.
- Run many trials.
- Record the proportion of successes.
- Use it to estimate the probability.
Example 1 — Simulate
How can you estimate the probability of heads by simulation?
Solution
Flip many times and take the proportion of heads.
| \(\hat p\) | \(=\) | \(\dfrac{\text{heads}}{\text{flips}}\) |
Example 2 — Law of large numbers
What happens to the proportion as trials increase?
Solution
It approaches the true probability.
| \(\text{proportion}\) | \(\to\) | \(P(\text{heads})=0.5\) |
Example 3 — Random assignment
Why randomly assign subjects to groups in an experiment?
Solution
Random assignment balances other factors, so differences can be attributed to the treatment.
Example 4 — Estimate from a trial
A spinner lands red \(18\) of \(60\) spins. Estimate \(P(\text{red})\).
Solution
Use the observed proportion.
| \(P(\text{red})\) | \(\approx\) | \(\dfrac{18}{60}=0.3\) |
Common pitfalls
Few trials give unstable estimates; use many.
Random assignment \(\neq\) random sampling — different purposes.
Short-run results vary even for a fair process.
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.
← Previous subtopic
Sampling and inference
Next subtopic →
Sample surveys, experiments, observational studies
More in Statistics and probability