--- title: "Experimental vs Theoretical" description: "Compare observed relative frequencies with a probability model. With independent trials and fixed outcome probabilities matching the model, larger samples usually give closer estimates. A particular l" canonical: https://lightmysky.com/learn/mathematics/experimental-vs-theoretical-mt_-bMnJcPJy8 source: https://lightmysky.com/learn/mathematics/experimental-vs-theoretical-mt_-bMnJcPJy8.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Experimental vs Theoretical Compare observed relative frequencies with a probability model. With independent trials and fixed outcome probabilities matching the model, larger samples usually give closer estimates. A particular longer run can still be farther away. Subject: Mathematics · Area: Probability · Ages 10 to 11 Page: https://lightmysky.com/learn/mathematics/experimental-vs-theoretical-mt_-bMnJcPJy8 ## Ready when they can - Roll a fair six-sided die 60 times and compare observed counts with the expected 10 per number - Explain why observed shares can differ from theoretical probabilities, can match them exactly, and tend to be closer in larger independent samples under the same model - Predict what would happen if the experiment were repeated 600 times instead of 60 ## Lesson: Experimental vs Theoretical Probability You want a die to treat everyone the same. A fair six-sided die has six equally likely faces, so the theoretical probability of a 3 is 1 in 6. Experimental probability is what you count: the number of 3s divided by the number of rolls. Probability times trials gives the expected count, so 60 rolls of a fair die are expected to give about 10 of each number. To judge a run, compare each observed count with the expected 10. *(drawing: Sixty rolls of a fair die: each number is expected about 10 times. These real counts land near 10, not exactly on it.)* **Example.** Zara flips a fair coin 10 times and gets 7 heads. That is not 50:50, but a fair coin can easily do that. With only 10 flips, results bounce around a lot. About half of a long run of fair flips is expected to be heads, so over 600 flips the share would very likely land much closer to half. With independent trials and a fixed chance, larger samples usually give shares closer to that chance. Independent means one result does not change the chance of another. A longer run can still be farther off. **Recap.** With independent trials and a fixed chance, larger samples usually give closer shares. A particular longer run can still be farther away. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Simple Chance Experiments](https://lightmysky.com/learn/mathematics/simple-chance-experiments-mt_J4j7d3iAfg) - [Calculating Simple Probability](https://lightmysky.com/learn/mathematics/calculating-simple-probability-mt_Zt30Gxi-qp) ## Opens up - [Experimental probability](https://lightmysky.com/learn/mathematics/experimental-probability-mt_sHJqh6UUya)