Repeated Experiments and Relative Frequency

A frequency is a ratio of collected counts

A fair coin gives heads probability one half on each independent trial. Start with twenty flips. The observed frequency is the number of heads divided by twenty, so its possible values are separated by one twentieth. This empirical fraction belongs to the collected data. It is not a replacement definition for the model probability.

Extend the same reproducible experiment from twenty to two hundred flips. Early outcomes stay in the total. One new observation now changes the fraction by much less than it did near the beginning. The trace can move either toward or away from one half at a particular step; convergence is not monotone improvement.

Read this experiment exactly

Begin with ten fair-coin trials using the fixed seed for this example. This particular sequence has two heads, so its heads frequency is two tenths, or twenty percent. The model probability remains fifty percent. The two solid theory bars and the experimental outlines describe different quantities.

Look at the right end of the running-frequency trace. It summarizes all ten results collected so far, not just the last flip. An endpoint below the gold probability line means the observed fraction of heads is below one half. It does not mean the next coin has become biased toward tails.

Extend the same sequence to twenty trials. There are now five heads in total, giving five twentieths, or one quarter. Three of the ten additional flips were heads. We added observations to the numerator and denominator together; we did not average the old fraction with the new count.

Continue to fifty trials. The accumulated number of heads reaches twenty-three, so the frequency is twenty-three fiftieths, or forty-six percent. This is closer to the model probability than the earlier twenty-five percent. It is a result of this particular prefix, not a promise of improvement at every step.

At one hundred trials, the sequence has forty-five heads. Its frequency is forty-five percent, slightly farther from fifty percent than the earlier forty-six percent. This finite example directly shows that collecting more data does not make every successive observed fraction closer to the model.

Convergence is not correction

This coin has a success probability of sixty-five percent. The observed proportion wanders as flips accumulate. Early results can be misleading because a single flip is a large fraction of a small sample. As the denominator grows, the proportion usually settles closer to the theoretical value. This is a statement about repeated independent trials with the same probability, not a promise that every finite experiment lands exactly on sixty-five percent.

A coin does not remember a run of tails and arrange compensating heads. The next success probability is still sixty-five percent under this independent model. Resetting the experiment clears collected trials; choosing a new seed makes a different reproducible trial sequence. Speed changes how quickly results are displayed, not the probability model. Try a short experiment several times, compare the variation, and resist interpreting a temporary imbalance as a new law.

Separate proportion error from count error

Keep the one-hundred-trial result visible. Relative to the model average of fifty heads, this sample is short by five heads. Dividing that count discrepancy by one hundred gives a proportion discrepancy of five percentage points. Count error and proportion error use different units.

Continue to two hundred trials. This sequence has ninety heads, compared with a model average of one hundred. The count discrepancy has doubled to ten, but ten divided by two hundred is still five percent. More observations can increase a count discrepancy without increasing its proportion discrepancy.

Continue to one thousand trials. There are four hundred seventy-three heads. That is twenty-seven below the model average of five hundred, but only two point seven percentage points below one half. The absolute count discrepancy grew again while the proportion moved closer to the model probability.

Travel along the running-frequency trace. Its horizontal coordinate counts the accumulated trials; its vertical coordinate is a fraction. The trace is not a running total of heads. A growing total can accompany either an upward or downward fraction, depending on what the new observations add.

For independent fair flips, the theoretical standard deviation of the heads count is one half times the square root of N. Dividing by N gives one over twice the square root of N for the proportion. Thus count spread grows while proportion spread shrinks. These are model spreads, not error bounds for this one trace.

Compare independent experimental sequences

Return to twenty trials, now using the second seed. This sequence has eight heads, giving forty percent, while the earlier seed gave five heads and twenty-five percent. Both use the same fair-coin model and the same sample size. The difference comes from the sampled outcomes.

The seed lets us reproduce an example exactly. Reopening this lesson should show the same eight heads at twenty trials. It is a tool for making the demonstration repeatable, not a parameter that assigns a different probability to heads. The probability control still reads one half.

Extend this second sequence to two hundred trials. It has ninety-seven heads, giving forty-eight point five percent. The first sequence at this size had ninety heads, or forty-five percent. Two experiments of equal length need not have identical errors, even when both use precisely the same model.

At one thousand trials, the second sequence has five hundred eight heads. Its observed fraction is fifty point eight percent, above the model value. The earlier thousand-trial sequence was below it. Finite sampling variation can occur on either side; the probability line is not a barrier that the trace cannot cross.

Do not treat successive prefixes of one trace as independent experiments. The first twenty trials are included in its first two hundred. Those observed fractions share data. To compare separate experimental runs, change the seed while keeping probability and sample size fixed, as we just did.

Compare experiments without changing the model

Keep success probability at sixty-five percent and collect two hundred trials using a different seed. The seed chooses a reproducible sequence; it does not change the probability assigned to the next trial. Comparing equal-length experiments makes finite-sample variation visible without quietly changing the question or the sample size.

Collect two thousand trials with that same seed. The initial two hundred outcomes are still the beginning of this sequence. Under independent identical trials, the standard deviation of the success proportion is the square root of p times one minus p divided by the number of trials. Increasing the denominator reduces this theoretical spread; no finite run guarantees exact equality.

Check a boundary where fluctuation disappears

Set success probability to one and begin again. Every trial succeeds, so the observed proportion is one as soon as there is a trial to count. The variance is zero. This boundary checks the distinction between a random-looking display and a probability model that can assign certainty.

Restore probability one half and use Interact for your own comparisons. Change the number of trials while holding probability fixed, then change probability while holding trial count fixed. State which change affects the mathematical model and which only provides more observations from it. A long-run tendency does not make a rare finite run impossible or make a coin compensate for its history.

Predict the effect of changing a model

Use a coin with heads probability one quarter. Before collecting data, its theory bars assign twenty-five percent to heads and seventy-five percent to tails. Equal-looking outcome symbols do not imply equal probabilities. We changed the model, so a frequency near one half would no longer be the natural long-run reference.

Collect one thousand trials from that quarter-probability model. The experimental outlines grow around a different pair of theoretical heights. The gold reference for heads is now one quarter. Compare frequency with the probability actually assigned here, rather than comparing every coin experiment with fifty percent.

Set heads probability to zero and collect one hundred trials. There are no heads: every trial is a tail. Once at least one trial exists, the observed heads fraction is exactly zero. This boundary has no sampling fluctuation because the model makes the chosen event impossible.

At probability one, every trial is a head instead. The observed heads fraction is exactly one for every positive sample size. Both boundaries agree with the spread formula, because p times one minus p is zero. A probability model can describe certainty as well as uncertainty.

Finally restore the fair model. Before pressing Interact, distinguish three experiments you could make: grow the same sequence, replace it with a different seed, or change the probability itself. The first changes sample size, the second changes the sampled data, and the third changes the model. Predict which displayed features should remain fixed in each case.