ACARA v9 CONTENT DESCRIPTION “conduct repeated chance experiments including those with and without equally likely outcomes, observe and record the results; use frequency to compare outcomes and estimate their likelihoods”
Builds on listing the possible outcomes of a chance experiment and deciding when outcomes are equally likely. Year 5 puts chance into action by repeating an experiment many times, recording every result, and using the frequencies to compare outcomes and estimate how likely they are. This works whether or not the outcomes are equally likely, and it shows how collecting data turns chance from a prediction into something that can be observed and measured.
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.
Repeating a chance experiment
Many trials reveal a pattern one trial cannot.
Toss the coin many times and record each result.
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.
Recording results as frequencies
The frequency is how many times an outcome happens.
Record each spin in a frequency table. Reveal the counts.
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.
Relative frequency
Relative frequency is the count divided by the total trials.
What share of the trials was blue in 50 spins?
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.
Comparing outcomes by frequency
The higher frequency is the more common result.
Which outcome has the larger frequency?
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.
Estimating likelihood from frequency
Frequent outcomes are estimated as more likely.
Use the frequencies to estimate the likelihood.
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.
Quick self-check
1. A repeated chance experiment means you...
2. The number of times an outcome happens is called its...
3. If heads came up 27 times in 50 tosses, its relative frequency is...
4. In 50 spins a spinner gave blue 30, red 12, green 8. The most likely colour is...
5. Doing many more trials usually makes a frequency estimate...