# Probability Units of Work

### Level 1 Probability

Achievement Objectives | Learning Outcomes | Unit title |

- use everyday language to talk about chance
- classify events as certain, possible, or impossible
| No Way Jose | |

Lonely Pig |

### Level 2 Probability

Achievement Objectives | Learning Outcomes | Unit title |

- use dice etc to assign roles and discuss the fairness of games
- play probability games and identify all possible outcomes
- compare and order the likelihood of simple events
| That's not fair | |

- recognise that not all things occur with the same likelihood
- observe some things are fairer than others
- explore to adjust the rules of games to make them fairer
| The Cube and Coin Challenge |

### Level 3 Probability

Achievement Objectives | Learning Outcomes | Unit title |

- make predictions based on data collected
- identify all possible outcomes of an event
- assign probabilities to simple events using fractions (1/2, 1/6 etc)
| What's in the Bag? | |

- determine an experimental estimate of the probability of simple events using frequency tables
- determine the theoretical probability of simple events using percentages, fractions and decimals
- systematically find all possible outcomes of an event using tree diagrams and organised lists
| I'm spinning | |

- systematically find all possible outcomes of an event using tree diagrams and organised lists
| Counting on Probability | |

- take samples and use them to make predictions
| Predict Away | |

- take samples and use them to make predictions
- compare theoretical and experimental probabilities
| Long Running | |

- make predictions based on the fraction of the spinner shaded
- compare theoretical and experimental probabilities
| Spinners |

### Level 4 Probability

Achievement Objectives | Learning Outcomes | Unit title | ||

- use relative frequency to predict events
- develop and evaluate strategies to win based on the relative frequency
- explain why probability is not an accurate predictor of events
| Top Drop | |||

- apply theoretical probabilities to solve problems
- use tree diagrams to find the possible outcomes for a sequence of events
- use powers of numbers
| Greedy Pig | |||

| - investigate probability in common situations;
- make and justify the probability of events in common situations;
- theoretically and experimentally examine the probabilities of games of chance.
| Beat It | ||

- find a theoretical probability
- use more than one way to find a theoretical probability
- check theoretical probabilities by trials
- identify what a fair game is and how to make a unfair game fair
| The Coloured Cube Question | |||

Heads and Tails | ||||

- use simulations to investigate probability in common situations
- predict the likelihood of outcomes on the basis of an experiment;
- petermine the theoretical probability of an event;
| Murphy's Law | |||

- theoretically and experimentally examine the probabilities of games of chance
- describe the notion of "short run variability"
- estimate and find the relative frequencies of events
| Gambling:who really wins? |

### Level 5 Probability

Achievement Objectives | Learning Outcomes | Unit title | ||

- explain what a probability distribution is
- explain how probabilities are distributed
- calculate what happens to the distribution when variables are summed
| Probability Distributions | |||

- Use long-run frequencies to estimate probabilities.
- Compare the results of theoretical and experimental approaches to games of chance.
| Fair Games |

### Level 6 Probability

Achievement Objectives | Learning Outcomes | Unit title |

- conduct straightforward experiments with coins, dice, spinners, and other random event generators
- produce and understand the concept of a random walk
- develop,understand and be able to use Pascal’s Triangle
- determine probabilities of a class that is just a little too complicated for probability trees.
| Investigating Random Processes |

Probability is the study of random events.

A 'random event' in probability is a collection of particular outcomes from a probability activity, for example, rolling a sum of 12 with two dice. These events are important both inside mathematics and outside it. The obvious examples to explore in a classroom setting involve games of chance, anything from a game of Ludo to a game of Roulette. On the other hand, businesses such as insurance companies need to know about events concerned with car accidents, death, and footballers having accidents. To decide whether it is worth taking some action, and what action to take, we rely on a measure of the likelihood of an event that we call **probability**. For practical reasons the size of the probability of an event is expressed as a number between zero and one or a percentage between zero and 100.

Probabilities can be determined in two ways: theoretically or experimentally. Many simple events can be found theoretically. For example, the probability of getting a 4, say, on the roll of a dice can be found theoretically. First you have to know the **event space**, the set of all possible outcomes. In the case of a dice this is {1, 2, 3, 4, 5, 6}. So there are six events in the event space. There is only one event in this event space that produces a 4. Hence the probability of getting a 4 when you roll a dice is 1/6, the 1 is for the number of times a four can occur among all possible events and the 6 is for the number of all possible events (the size of the event space). In general, when calculating probabilities theoretically, the probability is given by the equation

As another example consider what happens when you roll two dice. What are the chances of getting a sum of 4 then? Well here the event space is of size 36 {(1, 1), (1, 2), (1, 3), ... (6, 4), (6, 5), (6, 6)} and the times that 4 is possible are given by {(1, 3), (2, 2), (3, 1)}. Hence the probability of getting a 4 is 3/36 = 1/12.

But not all events can have their probabilities calculated theoretically. Think of the probability of there being three major earthquakes in a year or the chances that it will rain tomorrow. We can't produce an event space to measure these probabilities by. So we have to take measurements over a long period. In that time we can compare the number of rainy days over the number of days altogether - the number of favourable outcomes (yes, a rainy day is favourable in this context) over the number of possible outcomes. So in a case like this, where we have to calculate probability experimentally, the probability is given by