Use long-run frequency and evaluate a nonuniform probability model

Learning goal: Estimate probabilities from repeated trials, distinguish expected frequency from a guaranteed count and develop a model whose outcomes need not be equally likely. Compare a model with new observations and discuss possible discrepancies.

Relative frequency is the number of times an event occurs divided by the number of trials. If heads occurs 46 times in 80 tosses, its observed relative frequency is 46/80 = 0.575. This is a record of the trials, not proof that the device's exact probability is 0.575. A model probability describes the chance process; an observed proportion estimates it through data.

Use this explicitly invented cumulative coin record to examine variation: after 20 tosses there are 13 heads; after 80, 46; after 200, 102; after 500, 251; after 1,000, 508. The corresponding relative frequencies are 0.65, 0.575, 0.51, 0.502 and 0.508. These are simulated instructional values, not reports of a physical experiment performed for this lesson.

The record illustrates movement toward the neighborhood of one half without a perfectly monotonic approach. The last value, 0.508, is slightly farther from one half than 0.502. Long-run stabilization does not mean every added batch makes the proportion closer or that a short run must contain equal counts. An earlier excess of heads does not force tails to repair the record on the next independent toss.

Given a model probability p and n trials under stable conditions, np gives an expected or approximate count for planning. For a fair number cube, the event three or six has probability 2/6 = 1/3. In 600 rolls, about 200 occurrences are expected, but exactly 200 is not guaranteed. Keep the distinction between expected frequency and actual realized frequency visible.

Some chance processes do not have equally likely named outcomes. A tossed paper cup may land open end up, on its side or open end down. The physical shape and procedure can favor one orientation. Assigning one third to each because three labels exist would be an unsupported uniform model. Use observations to develop a model and retain the conditions under which they were collected.

Suppose an explicitly simulated first set of 100 cup trials records 60 open end up, 30 on the side and ten open end down. An empirical model assigns probabilities 0.60, 0.30 and 0.10, which sum to one. This model uses observed frequencies and may be approximate. It is not a universal claim about all paper cups or every tossing technique.

Under that model, a future 200 comparable trials would be expected to contain about 120 open-end-up results, 60 side results and twenty open-end-down results. The predictions are approximate counts. Conditions such as cup shape, release height, spin and surface should be kept similar when comparing observations with the model.

A separate simulated validation set of 100 gives counts 48, 38 and 14. Compared with the model's expected 60, 30 and ten, the differences are −12, +8 and +4. Possible explanations include chance variation, changed tossing conditions, recording errors or an inadequate model based on the first sample. The discrepancy is a reason to investigate, not immediate proof of one particular explanation.

More trials can help assess stability if the process remains comparable. Combining data from different procedures without noting the change can hide important differences. Before revising a model, check the records and methods, collect additional comparable trials and examine whether the discrepancy persists. A model should be updated because evidence supports a revision, not because one inconvenient result is discarded.

In an actual classroom experiment, define outcomes and ambiguous-case rules before starting. Use a light paper cup on a clear surface and record every valid trial. A digital simulation is another option if its outcome probabilities are stated; it tests consequences of those programmed probabilities rather than discovering the behavior of a physical cup. Label which kind of process produced the data.

Worked frequency: In 50 trials, an event occurs eighteen times. The estimate is 18/50 = 0.36. Using that provisional model for 200 similar trials predicts about 72 occurrences. It does not require the next event to occur or fail, and it does not guarantee the final count.

Worked cumulative record: Moving from 500 to 1,000 coin tosses in the supplied record adds 500 tosses and 257 heads, since 508 − 251 = 257. Do not add all cumulative head counts together; later totals already include the earlier trials.

Worked model comparison: The cup validation sample's open-end-up proportion is 0.48 compared with the initial model's 0.60. Report the twelve-percentage-point discrepancy and investigate conditions and sampling variation. Saying “the cup must be broken” selects a cause without supporting evidence.

Actual trial sequence: Choose a defined chance process and collect at least 50 trials using a consistent procedure, then pool comparable class trials when possible. Record cumulative relative frequency at several checkpoints. State the model probability if one is assumed, or build an empirical model from the observations if outcomes are not presumed equally likely. Keep this actual record separate from the lesson's illustrative values.

Independent performance: Present the outcome definitions, raw counts, relative frequencies, model and an approximate prediction for a larger number of comparable trials. Compare with a new set of observations and explain at least two plausible sources of discrepancy. Use a graph or table to show how relative frequency changes as trials accumulate, and explain why neither exact balancing nor monotonic improvement is guaranteed.