Probability Units of Work

Level 1 Probability

Achievement Objectives	Learning Outcomes	Unit title
S1-3	use everyday language to talk about chance classify events as certain, possible, or impossible	No Way Jose
S1-3		Lonely kiwi

Level 2 Probability

Achievement Objectives	Learning Outcomes	Unit title
S2-3	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
S2-3	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
S3-3	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 compare theoretical and experimental probabilities	Long Running

Level 4 Probability

Achievement Objectives

Learning Outcomes

Unit title

S4-3

S4-4

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

use simulations to investigate probability in common situations
predict the likelihood of outcomes on the basis of an experiment
determine the theoretical probability of an event

Murphy's Law

explore the theoretical and experimental probabilities of situations involving chance
recognise variability from theoretical expectations, especially with small numbers of trials
estimate and find the relative frequencies of events
distinguish between events and the outcomes that lead to those events
distinguish between dependent and independent events

What are the chances?

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 an unfair game fair.

Flip and roll

Level 5 Probability

Achievement Objectives

Learning Outcomes

Unit title

S5-3

S5-4

calculate the probabilities for outcomes of a dice roll.
draw a dot plot of results.
interpret dot plots in terms of centre and variation.
calculate averages and expected values.

Greedy Pig

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
S6-3	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