# Children’s understanding of probability

By Nuffield Foundation

## The Foundation has published Children’s Understanding of Probability, a literature review by Professors Peter Bryant and Terezinha Nunes from the University of Oxford.

In this review, the authors identify four ‘cognitive demands’ made on children when learning about probability and examine evidence in each of these areas: randomness, the sample space, comparing and quantifying probabilities, and correlations.

They draw together international evidence, from the early years through to adulthood, and highlight studies that are of particular relevance to teaching. They also identify areas that have been relatively neglected and would benefit from further research, particularly from fully evaluated intervention projects.

##### Randomness

Research using computer microworlds has shown that by the age of about ten, many children realise that there is an association between randomness and fairness, and that randomisation can be an effective way of ensuring fair allocations.

Both children and adults find it particularly difficult to understand the independence of successive events in a random situation. When tossing a coin for example, people often predict that a run of heads makes it more likely that the next toss will be tails (the ‘negative recency’ effect), or that the previous run of heads makes it more likely that heads will be the next result too (the ‘positive recency’ effect).

##### Sample space

Children can have particular difficulty with problems where some outcomes are equiprobable and others are not. For example, in throwing two dice at the same time, there are 36 possible equiprobable outcomes (1,1; 1,2; 1,3 etc.). But, if you record the result in terms of the sum of the two numbers thrown, there are only 11 possible outcomes and they are not equiprobable.

Understanding the sample space means children have to construct an exhaustive list of alternative, and uncertain, possibilities, but there is currently no research on their ability to do this.

##### Quantifying probability

Children understand proportions as ratios before they understand them as fractions, suggesting that children would learn about probabilities more easily if they are initially introduced as ratios.

Children and adults are much more likely to work out conditional probabilities correctly if the basic information is given as absolute numbers rather than as decimal fractions.

##### Correlations

Correlational thinking depends on children realising that the way to work out whether an association is random or not is to consider the relative amount of confirming and disconfirming evidence. When they use simple intuitive reasoning they often fall prey to a confirmation bias; they pay more attention to the confirming than to the disconfirming evidence.

##### Recommendations for further research

The authors make two main recommendations in relation to further research. Firstly that researchers should take advantage of research designs that have been successful in research on other aspects of children’s intellectual development. In particular, the combined use of intervention and longitudinal methods to study the links between the four aspects of probability.

Secondly, they recommend that more attention is paid to the great amount of related data that exists on other aspects of cognitive development. Probability makes a number of different cognitive demands and most of these demands are shared with other aspects of cognitive development about which we know a great deal. Probability is an intensive quantity, but so are density and temperature for example. Many people doing research on probability have not paid attention to research on these related topics, and have missed out on potentially valuable information.

##### Intervention study

The Foundation is now funding the authors to undertake a large-scale controlled intervention study of the teaching of probability to 9-to-10-year-olds.

## Related

By Nuffield Foundation

## Explore our projects

New

Education | 2020 - 2020

### The impact of COVID-19 on mainstream schools in England

View project
New

Education | 2020 - 2021

### Post-16 educational trajectories and social inequalities in political engagement

View project
New

Education | 2020 - 2022

### The effects of COVID-19 on families’ time-use and child development

View project
New

Education | Welfare | 2020 - 2021

### Growing up under COVID-19

View project
New

Education | 2020 - 2022

### Comparisons of cognitive skills and educational attainment across the UK

View project
In progress

Education | 2020 - 2022

### Competitive effects of free schools on student outcomes in neighbouring schools

View project
In progress

Education | 2020 - 2022

### Teacher supply, shortages and working conditions in England and Wales

View project
In progress

Education | 2020 - 2022

### The influence of headteachers on their schools

View project
In progress

Education | 2020 - 2021

### Teaching fractions and decimals to children aged 3 to 11

View project
In progress

Education | Welfare | 2020 - 2020

### Measuring the disadvantage attainment gap in 16-19 education

View project
New

Education | 2020 - 2021

### The ‘common elements’ approach: improving outcomes in early childhood education

View project
In progress

Education | 2019 - 2021

### Post-16 pathways: the role of peers, family background and expectations

View project
New

Education | 2020 - 2020

### The impact of COVID-19 on mainstream schools in England

View project
New

Education | 2020 - 2021

### The ‘common elements’ approach: improving outcomes in early childhood education

View project
New

Education | Welfare | 2020 - 2021

### Growing up under COVID-19

View project
New

Education | 2020 - 2022

### The effects of COVID-19 on families’ time-use and child development

View project
New

Education | 2020 - 2021

### Post-16 educational trajectories and social inequalities in political engagement

View project
New

Education | 2020 - 2022

### Comparisons of cognitive skills and educational attainment across the UK

View project
In progress

Education | 2020 - 2021

### Teaching fractions and decimals to children aged 3 to 11

View project
In progress

Education | 2020 - 2022

### The influence of headteachers on their schools

View project
In progress

Education | 2019 - 2020

### A systematic review of early years degrees and employment pathways

View project
In progress

Education | 2020 - 2022

### Teacher supply, shortages and working conditions in England and Wales

View project
In progress

Education | 2019 - 2022

### Do infants learn new words from educational picture books?

View project
In progress

Education | 2020 - 2022

### Competitive effects of free schools on student outcomes in neighbouring schools

View project
New

Education | 2020 - 2020

### The impact of COVID-19 on mainstream schools in England

View project
New

Education | 2020 - 2021

### Post-16 educational trajectories and social inequalities in political engagement

View project
New

Education | 2020 - 2022

### The effects of COVID-19 on families’ time-use and child development

View project
New

Education | Welfare | 2020 - 2021

### Growing up under COVID-19

View project
In progress

Education | 2020 - 2022

### Competitive effects of free schools on student outcomes in neighbouring schools

View project
New

Education | 2020 - 2022

### Comparisons of cognitive skills and educational attainment across the UK

View project
In progress

Education | Welfare | 2020 - 2020

### Measuring the disadvantage attainment gap in 16-19 education

View project
New

Education | 2020 - 2021

### The ‘common elements’ approach: improving outcomes in early childhood education

View project
In progress

Education | 2020 - 2022

### Teacher supply, shortages and working conditions in England and Wales

View project
In progress

Education | 2020 - 2022

### The influence of headteachers on their schools

View project
In progress

Education | 2020 - 2021

### Teaching fractions and decimals to children aged 3 to 11

View project
In progress

Education | 2019 - 2021

### Post-16 pathways: the role of peers, family background and expectations

View project
Reported

Education | 2019 - 2020

### A new mathematics GCSE curriculum for post-16 resit students

View project
Reported

Education | 2018 - 2018

### GCSE and A-level results day project

View project
Reported

Education | 2018 - 2020

### Teaching reading: an integrated programme for deaf & hearing children

View project
Reported

Education | 2018 - 2019

### Developing a sustainable intervention for disadvantaged children

View project
Reported

Education | 2017 - 2018

### Growing up digital

View project
Reported

Education | 2017 - 2020

### Mathematics in Further Education Colleges (MiFEC)

View project
Reported

Education | 2017 - 2020

### Educational attainment of children in need & children in care

View project
Reported

Education | 2017 - 2019

### Trialling an executive function training intervention for preschoolers

View project
Reported

Education | Welfare | 2017 - 2019

### Undermatch in higher education: prevalence, drivers and outcomes

View project
Reported

Education | 2017 - 2018

### The effect of retention and turnover on the teaching workforce

View project
Reported

Education | 2016 - 2017

### School choice and equality of opportunity

View project
Reported

Education | 2016 - 2018

### Modern Foreign Languages, phonics and reading strategies

View project
Search projects

We improve people’s lives by funding research that informs social policy, primarily in EducationWelfare and Justice. We also fund student programmes that give young people skills and confidence in science and research.

We offer our grant-holders the freedom to frame questions and enable new thinking. Our research must stand up to rigorous academic scrutiny, but we understand that to be successful in effecting change, it also needs to be relevant to people’s experience.