Key Stages 3 and 4: age 11-16

Links to the National Curriculum
Mathematics

• Collect data from a variety of suitable sources, including experiments.
• Interpret and discuss data
• Understand and use estimates of probability from theoretical models
• List all the outcomes for single events, and for two successive events in a systematic way.
• When dealing with a combination of two experiments, pupils identify all the outcomes, using diagrammatic, tabular or other forms of communication.
  

ICT

• Pupils use computer models of increasing complexity.

Duration
Two hours

Learning Objectives

Pupils will:

• Use the language associated with probability to discuss events including those with equally likely outcomes
• Compare experimental and theoretical probabilities in simple contexts.
• Understand that if an experiment is repeated there may be, and usually will be, different outcomes. Also that increasing the sample size leads to better estimates of probability
• Identify all the mutually exclusive outcomes of an experiment; know that the sum of probabilities is one and use this when solving problems.
• list all outcomes for single events, and for two successive events, in a systematic way: use a tree diagram to help to do this.
• Understand relative frequency as an estimate of probability and use this to compare outcomes of experiments.

Resources required
Coins, access to coins simulation

Learning activity

Dice and spinner games help develop the skills for understanding probability. These skills include:

• methods of organised counting
• comparing results of experiments to theoretical probabilities
  
• correctly using the language of probability

Tree diagrams are used as a systematic way of counting all possible outcomes for a simple event. Let’s consider the outcomes of tossing four coins.
List the possible outcomes for the first of the four coins
Draw branches from each of these outcomes. The number of branches will be the number of possible outcomes for the second coin.
Draw branches for each of the possible outcomes of tossing the third coin.
Read down the chart from the top to identify the 8 different combinations.

From the tree diagram above the eight possible outcomes are TTT, TTH, THT, HTT, THH, HTH, HHT, HHH. We can work out the probabilities of events related to the tossing of three coins. As there is only one outcome with three tails, the probability of getting three tails is 1/8. There are three outcomes with one head, the probability of getting one head is 3/8. You can probably see that the probability of getting at least one head is 7/8.

In all these examples we are using situations where a multiplicative rule of counting applies. This means that we can get the answer by multiplying two lots of numbers together. For instance to get the result of tossing two coins we multiply 2 by 2 as we can get two outcomes (T and H) on each toss. (To get the 8 outcomes from three tosses we multiply 2 by 2 by 2.)

This is because coin tossing in this way involves independent events even if we have one head on the first toss, we can still get a head on the second toss. The second toss does not rely on the first one in any way.

Use the coins simulation to compare the theoretical outcomes with experimental results.