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# Key Ideas: Key Ideas in Probability: Progression from Year 1-13

## Years 1-2 | NZC Level 1

The key idea of probability at NZC Level 1 is beginning to explore chance situations.

At Level 1 students are exploring games of chance that they play and outcomes from statistical investigations. They are discussing the different outcomes that are possible.

Students are developing early probability language and describing outcomes. For example an outcome is certain, possible, impossible or an outcome always happens, never happens, sometimes happens.

This key idea is extended in the key idea of probability at NZC Level 2 where students are beginning to recognise that some events are more likely than others in chance situations.

## Years 3-4 | NZC Level 2

The key idea of probability at Level 2 is beginning to recognise that some events are more likely than others in chance situations.

At Level 2 students are recognising the different likelihood of outcomes for games of chance that they play. They are describing outcomes from statistical investigations, for example, children in our class were more likely to have bananas in their lunch today than any other fruit (15/24 children had bananas in their lunch).

Students are developing more sophisticated nuances with probability language to describe outcomes, for example, more likely, less likely.

This key idea develops from the key idea of probability at NZC Level 1 where students were beginning to explore chance situations.

This key idea is extended in the key idea of probability at NZC Level 3 where students are quantifying one-stage chance situations by deriving probabilities and probability distributions from theoretical models and/or estimating probabilities and probability distributions from experiments.

## Yers 5-6 | NZC 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 chance situations which can arise:

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 ½ 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. (Try it! Another example is rolling a 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 NZC 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 NZC 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.

## Years 7-8 | NZC Level 4

The key idea of probability at Level 4 is estimating probabilities and probability distributions from experiments and deriving probabilities and probability distributions from theoretical models for two-stage chance situations.

At Level 4 students are experimenting with two-stage chance situations, for example tossing two coins, dice, pigs, drawing pins, a coin and a die. Students are systematically listing all possible outcomes including using tools such as two way tables. Students are recording their results and plotting frequencies of outcomes. Students are recognising that in some chance situations outcomes are not equally likely, for example getting two heads or a head and a tail.

Students are comparing their experimental results with others in the class. They should also where possible compare estimated probabilities and probability distributions from experiments with theoretical model probabilities and probability distributions.

Starting with a one-stage chance situation (e.g., tossing one coin) students need to recognise that estimates of probabilities from experiments should be based on a very large number of trials. Technology is a useful tool for large numbers of trials.

In addition students should be exploring the idea of independence in probability using a one-stage chance situation, for example tossing one coin or one pig. In this setting independence refers to sequences of trials where the outcome of one trial has no impact on the next trial. For example, five tosses of a fair coin have come up HHHHH, the probability of getting a head on the next toss is still ½.

The three different types of chance situations described in NZC Level 3 need to be reinforced at this and other levels.

Link to statistical investigations: Students are exploring outcomes for two simple categorical variables in statistical investigations from a probabilistic perspective. For example, using gender and tongue roll or not, what is the probability that a randomly selected student is a boy and can roll his tongue?

This key idea develops from the key idea of probability at NZC Level 3 where students are quantifying one-stage chance situations by deriving probabilities and probability distributions from theoretical models and/or estimating probabilities and probability distributions from experiments.

This key idea is extended in the key idea of probability at NZC 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

## Years 9-10 | NZC Level 5

The key idea of probability at Level 5 is 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.

At Level 5 students are exploring two- and three-stage chance situations, for example using spinners, drawing marbles out of a bag (with replacement) tossing three dice, or rock, scissors and paper. More complex chance situations can be explored, for example two spinners with different colour combinations on them, or two bags of marbles with different numbers of each colour in them, or a mixture such as a bag of marbles and a spinner. Problems include ones like “Is the game fair?”

Students are systematically listing all possible outcomes using tree diagrams and other relevant tools such as two way tables. Students are recording their results and plotting frequencies of outcomes. Students are aware that in some chance situations outcomes are not equally likely.

Students are where possible comparing experimental estimates with theoretical model probabilities. Students need to recognise that theoretical model probabilities and experimental estimates of probabilities are approximations of the true probabilities which are never known. All probabilities lie in the range 0-1.

The three different types of chance situations described in NZC Level 3 need to be reinforced at this and other levels.

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

This key idea develops from the key idea of probability at NZC 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.

This key idea is extended in the key idea of probability at NZC Level 6 where students are exploring chance situations involving discrete random variables.

## Years 11 | NZC Level 6

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 NZC Level 3 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 NZC 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 NZC Level 7 where students are investigating chance situations involving continuous variables and using more sophisticated tools and ideas.

## Year 12 | NZC Level 7

The key idea of probability at Level 7 is investigating chance situations involving continuous variables and using more sophisticated tools and ideas.

At Level 7 students are exploring chance situations involving continuous variables, for example, using a technology experiment to generate an estimated probability distribution of the angle of the pointer from north for a spinner (uniform distribution) or a physical experiment (rice bombing) to generate an estimated probability distribution of the absolute horizontal distance from the target point’s line (normal distribution).

Students are investigating chance situations using tools such as probability tree diagrams, two way tables, simulations and technology.

The three different types of chance situations described in NZC Level 3 need to be reinforced at this and other levels.

Link to statistical investigations: Students are using results from statistical investigations to explore and ascertain risk and relative risk and are interpreting and communicating risk ideas.

This key idea develops from the key idea of probability at NZC Level 6 where students are exploring chance situations involving discrete random variables.

This key idea is extended in the key idea of probability at NZC Level 8 where students are investigating chance situations using probability concepts and distributions.

## Year 13 | NZC Level 8

The key idea of probability at Level 8 is investigating chance situations using probability concepts and distributions.

At Level 8 students are investigating chance situations using concepts such as randomness, probabilities of combined events and mutually exclusive events, independence, conditional probabilities and expected values and standard deviations of discrete random variables, and probability distributions including the Poisson, binomial and normal distributions.

Students need to be able to model chance situations using both discrete and continuous probability distributions and apply probability concepts. They are using theoretical model probabilities and estimating probabilities from experiments as appropriate.

The three different types of chance situations described in NZC Level 3 need to be reinforced at this and other levels.

This key idea develops from the key idea of probability at NZC Level 7 where students are investigating chance situations involving continuous variables and using more sophisticated tools and ideas.

Source: http://www.nzmaths.co.nz/key-mathematical-ideas