# Probability: Level 6

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The key idea of probability at level 6 is exploring chance situations involving discrete random variables.

At level 6 students are exploring discrete chance situations.  For example, students could explore the distribution of the number of girls in a five child family.  The probability of having a girl is ½. The possible outcomes are 0, 1, 2, 3, 4, or 5 girls.  They could derive theoretical model probabilities by systematically listing all the different ways they could get 0, 1, 2, 3, 4, or 5 girls, and/or they could estimate probabilities via experiment using technology or other means.  Alternatively they could explore the distribution of the number of successful hoops in three free throws in basketball.  In this situation they would need to physically shoot the hoops and estimate the probabilities. This is laying the foundations for the binomial distribution in level 8.

Another example which can be explored both ways is the situation where an imaginary two-eyed being has a combination of red, green and yellow eyes.  The colour for each eye is equally likely to be red, green or yellow and the colour of one eye is independent of the other eye.  Possible explorations include looking at probabilities (theoretical model and/or estimated via experiment) of the outcomes for 0, 1, or 2 red eyes (binomial) and/or probabilities of given combinations of eye colour, for example yellow/green eyes.

Students need to appreciate the role sample size plays in estimating probabilities via experiment.   Students are recording their results and plotting frequencies of outcomes and starting to get a sense of probability distributions.  Students are aware that in some chance situations outcomes are not equally likely.

The three different types of chance situations described in level three need to be reinforced at this and other levels.

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

This key idea develops from the key idea of probability at level 5 where students are estimating probabilities and probability distributions from experiments and deriving probabilities and probability distributions from theoretical models for two- and three-stage chance situations and recognising the connections between experimental estimates, theoretical model probabilities and true probabilities.

This key idea is extended in the key idea of probability at level 7 where students are investigating chance situations involving continuous variables and using more sophisticated tools and ideas.