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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
With independent trials and a fixed chance, larger samples usually give closer shares. A particular longer run can still be farther away.
Rewritten readings were written for this dataset.
More to read
Theoretical and experimental probability: Coin flips and die rolls
Theoretical probability comes from counting equal chances: a fair die has six faces, so each number has probability 1 in 6. In 60 rolls you therefore expect about 10 of each number (60 divided by 6). Expected is a prediction, not a promise. Experimental probability comes from actually doing it: count what happened and divide by the total rolls. Twelve 6s in 60 rolls is an experimental share of 12 in 60. Small experiments wobble around the theory, and that wobble is normal, not proof of cheating.
Compare dice results with predictions
Roll one fair six-sided die repeatedly, with each roll independent, and tally the faces. Each face has a theoretical chance of 1/6. Across 60 rolls, the expected count is 10 for each face. Actual counts vary, although they can match exactly.
With more rolls, the proportions usually move closer to the predicted chances. For example, 96 to 104 sixes in 600 rolls means about 16% to 17.3%. That range is closer to 1/6 than 8 to 12 sixes in 60 rolls. Larger samples tend to reduce the effect of lucky streaks on proportions, not necessarily on differences in counts. Any individual run can still stray from the prediction.
Predicted chances and observed shares
A sample space lists possible results: six faces for a die, two sides for a coin, or four colours for a four-colour spinner. When outcomes are equally likely, divide the number you want by the total number. Rolling a 4 with a fair die has probability 1/6.
For independent trials with unchanged chances that fit the model, observed proportions tend towards theoretical probabilities. You might get twelve 6s in sixty rolls. Six hundred rolls will usually give a proportion closer to 1/6, but a longer run can have a larger proportional error. Judge repeated patterns, rather than a few memorable results.
Plan a fair dice investigation
Before gathering results, choose one die and a consistent way to roll and tally it. A class can try sixty rolls. Record every outcome; selecting only interesting rolls distorts the results.
Then compare the counts with their expected values. Ask whether differences could reflect chance or a biased die. Getting 12 of a face instead of the expected 10 can happen by chance. Exactly 10 of every face is possible too and does not by itself show a problem. Plan first, record next, and draw conclusions afterwards.
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2 · Watch
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18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.