# Probability: Level 3

The key idea of probability at level 3 is quantifying one-stage chance situations by deriving probabilities and probability distributions from theoretical models and/or estimating probabilities and probability distributions from experiments.

At level 3 students are beginning to explore one-stage chance situations, for example tossing a drawing pin, throwing a die from a board game, tossing or spinning a coin, rolling a pencil, or tossing a “pass the pigs” pig.  They are listing possible outcomes for situations.  Students are recording their results and plotting frequencies of outcomes.  Students are comparing their experimental results with others in the class, acknowledging that results may vary.  Students are recognising that in some chance situations outcomes are not equally likely, for example tossing a pig or drawing pin.

Students are developing an awareness that some types of chance situation can be easily modeled (for example tossing a coin, where there are two equally likely outcomes), while others are more complex and can only be modeled in the classroom by estimating probabilities based on experiments (for example tossing a drawing pin).

Students are learning about the three different types of model which can arise in chance situations.

1. Good model: An example of this is the standard theoretical model for a fair coin toss where heads and tails are equally likely with probability 1/2 each. Repeated tosses of a fair coin can be used to estimate the probabilities of heads and tails. For a fair coin we would expect these estimates to be close to the theoretical model probabilities.
2. No model: In this situation there is no obvious theoretical model, for example, a drawing pin toss. Here we can only estimate the probabilities and probability distributions via experiment. (These estimates can be used as a basis for building a theoretical model.)
3. Poor model: In some situations, however, such as spinning a coin, we might think that the obvious theoretical model was equally likely outcomes for heads and tails but estimates of the outcome probabilities from sufficiently large experiments will show that this is a surprisingly poor model. (Another example is rolling a hexagonal pencil.) There is now a need to find a better model using the estimates from the experiments.

Link to statistical investigations: Students are exploring outcomes for single categorical variables in statistical investigations from a probabilistic perspective.

This key idea develops from the key idea of probability at level 2 where students are beginning to recognise that some events are more likely than others in chance situations.

This key idea is extended in the key idea of probability at level 4 where students are estimating probabilities and probability distributions from experiments and deriving probabilities and probability distributions from theoretical models for two-stage chance situations.