ACARA v9 CONTENT DESCRIPTION “conduct repeated chance experiments and simulations, using digital tools to determine probabilities for compound events, and describe results”
Compound events
A compound event is one made of more than one part. Getting two heads when two coins are tossed is a compound event, and so is rolling a total of 7 on two dice, because each depends on more than a single result. We describe a compound event by combining its parts, then ask how likely the whole combination is. This unit keeps to one running example, two coins and the event two heads, and works out its probability two different ways so the two ways can be compared.
A compound event
Two coins give four outcomes; two heads is one of them.
a compound event has more than one part; here two heads in two tosses, just one of four outcomes.
Theoretical probability
Theory finds a probability by counting the equally likely outcomes, exactly as a table or tree does. When two coins are tossed there are four equally likely outcomes, written HH, HT, TH and TT. Just one of these four is two heads, so the theoretical probability of two heads is 1/4, which is 0.25. Nothing has been tossed yet; this value comes purely from the model of the coins and from counting. It is the benchmark that every experiment in this unit will be measured against.
Theoretical probability
Count the equally likely outcomes from the model.
theory counts the equally likely outcomes: one of four is two heads, so the theoretical probability is 1/4 = 0.25.
Experimental probability
An experimental probability, also called a relative frequency, counts what actually happens. Toss the two coins many times, count how often two heads appear, and divide that count by the number of trials. In a short run the result wanders: twenty trials might give six two-head results, an estimate of 6/20, or 0.30, which sits above the theoretical 0.25 purely by chance. A different twenty trials could land below it. A small experiment rarely matches the theory exactly, and on its own it is not yet a reliable guide.
Experimental probability
A short run of 20 paired tosses.
experiment counts what actually happened: in a short run the relative frequency can differ from the theory.
Running a simulation
Carrying out thousands of tosses by hand would take far too long, so we use a simulation. A simulation imitates the situation with a quick model, such as coins, dice, or a random number tool, and repeats it as many times as we like in moments. Each single run records only whether the compound event happened, here whether the pair came up two heads, and the many results are tallied into a relative frequency. The model stands in for the real coins, so the counting can be done at speed.
A simulation settling
The estimate steadies as the trials grow.
a simulation repeats the experiment many times; as trials grow, the estimate settles toward 0.25.
Comparing experiment and theory
As the number of trials grows, the experimental estimate tends to settle toward the theoretical value. This is the same sample-size effect met in statistics: more trials steady the estimate. After ten or fifty trials the running estimate of two heads may still wobble well away from 0.25, but by a thousand trials it usually lands close, perhaps about 0.247. The two numbers come close yet are rarely exactly equal, and that small gap is expected. A large gap, however, would be a warning that the model or the counting has gone wrong.
Experiment beside theory
Close after many trials, rarely exact.
with many trials the experimental estimate comes close to the theoretical 0.25, though rarely exactly equal.
Why this matters
Simulation is how people tackle real problems that are too hard to count exactly, in science, engineering, finance and games: build a model, run many trials, and read off the relative frequency. Comparing that experimental result with theory checks both the model and the mathematics at once, and it reminds us that probability describes the long run, not any single try. The common slip is to expect a short run to match the theory exactly, or to trust an estimate built from too few trials. This completes the Year 8 probability units.
Quick self-check
1. What is a compound event?
2. Tossing two fair coins, what is the theoretical probability of two heads?
3. How is an experimental probability found?
4. What happens to the experimental estimate as the number of trials grows?
5. After many trials, how should the experimental estimate compare with the theoretical 0.25?