Repeated Experiments
Repeating a chance experiment
A single toss of a coin or roll of a die tells very little, because chance makes each result unpredictable. Repeating the experiment many times is what reveals a pattern. A repeated chance experiment means carrying out the same trial over and over, under the same conditions, and recording the outcome each time. Tossing a coin fifty times, or spinning a spinner one hundred times, builds up a record of results. The more trials are done, the clearer the pattern in the outcomes becomes, which is why repetition is at the heart of studying chance with data.
Recording results as frequencies
As an experiment is repeated, the results must be recorded carefully, usually with tally marks gathered into a frequency table. The frequency of an outcome is simply the number of times it happened. After fifty spins of a four-colour spinner, the table might show blue twenty-one times, red fourteen, green nine and yellow six. Recording every trial and counting the frequencies turns a stream of single results into an organised summary. A good record is the foundation for everything that follows, because all the comparing and estimating is done from these frequencies.
Relative frequency
A raw frequency is more useful when compared to the total number of trials, giving the relative frequency. The relative frequency of an outcome is its frequency divided by the number of trials, written as a fraction. If heads came up twenty-seven times in fifty tosses, its relative frequency is twenty-seven out of fifty. Relative frequency expresses how large a share of all the trials an outcome took, which makes results from experiments of different sizes comparable. It is the bridge between the count of an outcome and an estimate of how likely that outcome is.
Comparing outcomes by frequency
Frequencies make it easy to compare outcomes. The outcome that happened most often has the highest frequency, and the one that happened least has the lowest. In fifty spins where blue came up far more than the other colours, blue is clearly the most common result. Comparing the frequencies, or the relative frequencies, shows at a glance which outcomes occur more and which occur less. This comparison works for any experiment, including one where the outcomes are not equally likely, because the data show what actually happened rather than what was assumed.
Estimating likelihood from frequency
The frequencies from a repeated experiment can be used to estimate how likely each outcome is. An outcome that occurs often in many trials is estimated to be more likely, and one that occurs rarely is estimated to be less likely. If a spinner lands on blue thirty times out of fifty, blue is estimated as the most likely colour, and the spinner is probably not fair. The estimate improves as the number of trials grows, because more data give a steadier picture. Estimating likelihood from frequency is how chance is measured when the outcomes cannot simply be assumed to be equally likely.
Letting the data speak
Studying chance with experiments means letting the data speak: repeat the trial many times, record each result as a frequency, turn frequencies into relative frequencies, compare the outcomes, and estimate their likelihoods. The results of a few trials can be misleading, but a large number of trials gives a reliable guide, even when the outcomes are not equally likely. With these habits a child can run a chance experiment, organise its results, and draw a sensible conclusion about how likely each outcome is.