GCSE OCR GCSE (91) Maths
Section 11  Probability
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Introduction
Overview
Delivery guides are designed to represent a body of knowledge about teaching a particular topic and contain:
 Content: a clear outline of the content covered by the delivery guide;
 Thinking Conceptually: expert guidance on the key concepts involved, common difficulties learners may have, approaches to teaching that can help learners understand these concepts and how this topic links conceptually to other areas of the subject;
 Thinking Contextually: a range of suggested teaching activities using a variety of themes so that different activities can be selected that best suit particular classes, learning styles or teaching approaches.
Curriculum Content
Overview
GCSE content Ref.  Subject content  Initial learning for this qualification will enable learners to…  Foundation tier learners should also be able to…  Higher tier learners should additionally be able to…  DfE ref. 

OCR 11  Probability  
11.01  Basic probability and experiments  
11.01a  The probability scale  Use the 01 probability scale as a measure of likelihood of random events, for example, ‘impossible’ with 0, ‘evens’ with 0.5, ‘certain’ with 1.  P3  
11.01b  Relative frequency  Record, describe and analyse the relative frequency of outcomes of repeated experiments using tables and frequency trees.
 P1
 
11.01c  Relative frequency and probability  Use relative frequency as an estimate of probability.
 Understand that relative frequencies approach the theoretical probability as the number of trials increases.  P3,
P5  
11.01d  Equally likely outcomes and probability  Calculate probabilities, expressed as fractions or decimals, in simple experiments with equally likely outcomes, for example flipping coins, rolling dice, etc. Apply ideas of randomness and fairness in simple experiments. Calculate probabilities of simple combined events, for example rolling two dice and looking at the totals. Use probabilities to calculate the number of expected outcomes in repeated experiments.  P2,
P7  
11.02  Combined events and probability diagrams  
11.02a  Sample spaces  Use tables and grids to list the outcomes of single events and simple combinations of events, and to calculate theoretical probabilities. e.g. Flipping two coins.
 Use sample spaces for more complex combinations of events. e.g. Recording the outcomes for sum of two dice.
 Recognise when a sample space is the most appropriate form to use when solving a complex probability problem. Use the most appropriate diagrams to solve unstructured questions where the route to the solution is less obvious.  N5,
P6, P7 
11.02b  Enumeration  Use systematic listing strategies.  Use the product rule for counting numbers of outcomes of combined events.  N5  
11.02c  Venn diagrams and sets  Use a twocircle Venn diagram to enumerate sets, and use this to calculate related probabilities. Use simple set notation to describe simple sets of numbers or objects. e.g. A = {even numbers} B = {mathematics learners} C = {isosceles triangles}  Construct a Venn diagram to classify outcomes and calculate probabilities. Use set notation to describe a set of numbers or objects. e.g. D = {x : 1 < x < 3} E = {x : x is a factor of 280}  Construct tree diagrams, twoway tables or Venn diagrams to solve more complex probability problems (including conditional probabilities; structure for diagrams may not be given).  P6, P9

11.02d  Tree diagrams  Use tree diagrams to enumerate sets and to record the probabilities of successive events (tree frames may be given and in some cases will be partly completed).  P6, P9
 
11.02e  The addition law of probability  Use the addition law for mutually exclusive events. Use p(A) + p(not A) = 1  Derive or informally understand and apply the formula
 P4
 
11.02f  The multiplication law of probability and conditional probability 
 Use tree diagrams and other representations to calculate the probability of independent and dependent combined events.  Understand the concept of conditional probability, and calculate it from first principles in known contexts. e.g. In a random cut of a pack of 52 cards, calculate the probability of drawing a diamond, given a red card is drawn. Derive or informally understand and apply the formula
Know that events A and B are independent if and only if
 P8,
P9 
This delivery guide provides support for the delivery of the following topic areas from the curriculum content:
 the probability scale
 relative frequency and probability
 equally likely outcomes and probability
 sample spaces
 enumeration
 tree diagrams and Venn diagrams
 addition law
 multiplication law and conditional probability.
The following activities are designed to give learners practice in each of the basic skills.
[11.01a] Ordering Probabilities Washing Line
Use a washing line, length of string or a rope to represent the probability scale. Give learners some initial event cards that are meaningful to them to place on the scale between 0 and 1.
This could be extended to a discussion of possible events that would be placed at specific positions on the line.
This activity forms an introduction to randomness and can promote discussion about events.
[11.01b] Buffon’s Needle (Interactivate)
[11.01c] Adjustable Spinner (Interactivate)
[11.01d] Probability Worksheet (TES)
[11.02a] Sample Space Diagrams (TES)
[11.02b] Systematic List (MathCatchers)
Three listing puzzles which encourage learners to use a systematic approach to find all the possible solutions.
The Baffling Bicycle Lock and Pizza Pitas problems do not require long initial teacher explanations, but the Tooth Fairy Problem relies on learners knowing, or being told, the values of US coins.
[11.02c] Venn Diagrams (NRICH)
[11.02d] Probability (Bitesize)
[11.02e] Mutually Exclusive Events (Interactive Mathematics)
[11.02f] Conditional Probability (TES)
Thinking Conceptually
Overview
Approaches to teaching the content
Probability allows for many reallife applications of mathematics. Most learners will have an idea of randomness, and there is plenty of opportunity for experimentation here. Learners should be given the opportunity to work with dice, coins, cards, etc. to demonstrate randomness. Plenty of experimentation is necessary to ensure a good understanding; learners can collect their own data and then make inferences from this data. They can try to make predictions and begin to develop an understanding that probability uses existing evidence to make predictions about what may happen in the future. The National Lottery offers a nice starting point; most learners know someone who enters and it allows for discussion of the likelihood of winning some of the different amounts.
Common misconceptions or difficulties learners may have
Learners can have difficulties understanding randomness as a concept; the use of dice and coins may help this.
Learners can have difficulty understanding that not all events are equally likely, e.g. “I have bought my lottery ticket so now my chance of winning is 5050”.
Learners can sometimes have difficulty understanding independence of events and think that later events are affected by what has already happened e.g. “I have flipped a coin and got a heads three times so the next time it must be a tails”.
Conceptual links to other areas of the specification – useful ways to approach this topic to set learners up for topics later in the course
This topic area links to working with fractions, decimals and percentages and it would benefit learners understanding if teachers plan for some time to be spent developing confidence in working with each numerical form and their equivalence. Probability questions may also link to number properties, ratios and geometric shapes.
There is a lot of problem solving involved in probability and learners will have to be able to extract information in order to determine solutions. Teachers should ensure that time is spent on these longer questions so that learners have experience of extracting the relevant information from the problem. It is strongly suggested that, where possible, teachers provide real examples to emphasise the importance of probability in realworld contexts; examples could include medical statistics, games of chance, and insurance.
Probing Questions
Use the following examples to assess learners' understanding:
1. Which of the following are TRUE or FALSE?
 Experimental probability is more reliable than theoretical probability
 Relative frequency finds the true probability
 Experimental probability gets closer to the theoretical probability if more trials are carried out
2. Fred rolls a dice 20 times and records the number of times he obtains each number. He gets 8 sixes. Do you think the dice is fair? Explain your answer.
3. Convince me that the probability of getting 2 heads from 2 coins is ¼.
4. Show me a pair of mutually exclusive events.
5. Explain what we mean by independent events.
6. Two coins are thrown at the same time. There are four possible outcomes: HH, HT, TH, TT. How many possible outcomes are there if three coins are used? Or four coins are used? Or five coins are used?
[11] Probability Through Problems: A New Approach to Teaching Probability (NRICH)
An article and collection of resources with some suggestions for approaching probability.
This includes some contextual resources, also included in the Thinking Contextually section of this Delivery Guide.
[11] Thoughts and Crosses (TES)
[11.01d] Evaluating Statements about Probability (MARS Shell Centre)
This is a full lesson plan with prelearning activity. All lesson materials and assessments are included.
The lesson is designed to help address common misconceptions about probability and to assess learners understanding of equally likely events, randomness and sample sizes.
The activities encourage learners to challenge their preconceptions and explain their work.
It draws on prior knowledge and extends their understanding through class discussions.
[11.01d] Derren Brown 10 Heads in a row (NRICH and YouTube)
Derren Brown claimed to have flipped a coin 10 times and got 10 heads in a row.
The video shows how he did it and the activity allows for some thoughtprovoking questions and discussions on likelihood.
[11.02c] Venn Diagrams (TES)
This resource has two activities which check learners’ understanding of Venn diagrams.
One is a mystery activity where learners have to annotate a Venn diagram using clues on cards and then calculate a probability.
The second can be used as an extension activity.
[11.02c] Representations: General Analysis (NRICH)
Thinking Contextually
Overview
Approaches to teaching the content
A major advantage of teaching probability is that there is a very reallife context within which the topic exists. It is suggested that wherever possible teachers aim to ensure that the context is accessible in order to avoid confusion and that the context interests the learners. It can also help with understanding if the results and inferences are not obvious as this can also help to maintain interest.
Some simple, well known contexts are given here:
Sport – Most competitions involve an initial flip of a coin to determine who starts the match, a “5050” chance. Based on a basketball player’s past performance the chance of them scoring from a single shot can be determined. In football, the penalty taker and the goalkeeper often use past experiences to inform the direction they choose.
Board Games – A spinner has four equalsized sections (black, red, white and blue) so there is a 25% chance of it landing on red. Similarly the odds of rolling a fair, sixsided dice and getting an even number is 50%, since three of the six numbers on the dice are even.
Medical Decisions – When a patient is advised to undergo surgery they often want to know the success rate of the operation. Again, this is just a probability.
Weather – When planning an outdoor activity, people generally check the probability of rain. Meteorologists also predict the weather based on the patterns of the previous year. Temperatures and natural disasters are also predicted using probability; they are not stated as definite, but as approximate chances.
[11.01c] Level 2 Data Handling (Nuffield Foundation)
[11.01d] Design a Board Game (Shell Centre for Mathematical Education)
[11.02f] Conditional Probability (NRICH)
An article from NRICH with some questions and scenarios to get learners thinking. It also links to Who Is Cheating? and The Dog Ate My Homework.
These two tasks are well differentiated and allow for lots of discussion in class about theoretical probabilities applied to a real context.