Children’s understanding of probability

Dealing with uncertainty and probability

When events happen randomly, we cannot be certain about what will happen next.

But we can analyse and compare the probabilities of particular events logically and mathematically, provided that we know enough about all the possible things that could happen (the sample space)

Of course this kind of analysis makes a variety of intellectual demands on the people carrying it out.

The questions that we tried to answer in our report were: (a) what these demands are (b) how to help children learn to satisfy them.

Conclusions of the report on children’s understanding of probability

Learning about probability makes three main cognitive demands:Understanding the nature and the value of randomness and how it

leads to uncertaintyIdentifying and working out the sample space (all the possible

outcomes)Quantifying and comparing probabilities: a proportional task There is a great deal of good research on children learning these

aspects of probability, but much of this was done in other (non-probability) contexts

This research should be the basis for devising new ways of teaching probability

1. The nature of randomness and randomisation

Randomness & randomisation: the problems and the solutions

Piaget & Inhelder claimed that young children can’t discriminate random from determined sequences

Problem of reversibility vs irreversibility

They worked with 5-13yr-olds on progressive randomisation

Younger children predictedcontinued order

Older children predictedprogressive mixing

Mixed vs ordered Children may do a great deal better than they do in the tilting

tray task if they are given randomisation tasks in more familiar contexts: like shuffling cards

It is probably the case that older children and adults get the idea of randomisation leading to mixed outcomes altogether too well

This is apparent in the “representativeness error” described by Kahneman & Tversky:

Many adults judge the order of the next six babies as more likely to be BGGBGB than BBBGGG, but the probability of both is the same . P=016 (1/64 )

The independence of successive events in random sequences

One hallmark of random sequences is the independence of successive events

Many adults either forget this or do not understand it when they make the very common negative recency error (lightning never strikes twice).

What about children?

A purple marble

A yellow marble

Each colour is just as likely

15 yellow marbles

15 purple marbles

15 purple and 15 yellow balls in a bag Someone has already drawn four balls from the bag

(replacing the ball after each draw) and all four were purple. This person is going to make another draw. What is likely to happen on his next draw?

1. The next draw is more likely to be a purple ball than a yellow one; 2. The next draw is more likely to be a yellow ball than a purple one; 3. The two colours are equally likely.

Positive recency

Negative recency

Correct answerChiesi & Primi

Difficulty in understanding the independence of random events

Percentages for the different answers in Chiesi & Primi’s task

Positive Negative Correct

recency recency answer

8yrs 0

10yrs 40

College 41

student

Percentages for the different answers in Chiesi & Primi’s task

Positive Negative Correct

recency recency answer

8yrs 66 34 0

10yrs 30 30 40

College 16 43 41

student

The social value of randomness

Randomisation is one way of ensuring fairness, as in a lottery or in other forms of selection (e.g. for children's games).

It is also a necessary part of starting some games (shuffling cards, throwing a coin) to ensure that one team or competitor does start with an unfair advantage.

We really need to study children’s understanding of randomeness in this sort of familiar context

Uses of randomisation: fairness

Paparistodemou et al worked with 5-8yr-old children on a computer microworld called Space Kid with a game about fair distribution

In this game the space kid is in peril of hitting a blue mine below him and a red mine above him

His up-down movements are determined by the number of times that a white ball moving unpredictably around hits a number of red and blue balls

Paparistodemou, Noss &Pratt

Paparistodemou, Noss &Pratt

Comments on the Paparistodemou et al. study

Making the connection between fairness and randomisation (‘unsteerable fairness’) is an excellent idea

There are other ways of achieving fairness: one is a controlled predictable procedure like sharing.

We need to compare tasks in which randomisation is a better way of achieving fairness than sharing (e.g. lotteries) and other tasks in which sharing is a better way than randomisation

Conclusions on randomness and randomisation

Young children may have some difficulties in distinguishing the nature of non-random, determined events from random, uncertain events

However, probably through informal experience, they do seem to take quite easily to the idea of a link between randomisation and fairness

This link should provide a good way of teaching children about randomisation and uncertainty

2. Sample space

HHHHHHTHHTHHHTTHTTTHTTHHTHTHTHHH

HHHTHHTTHTHTHTTTTTTTTTHTTHTTTHHT

The sample space of four successive tosses of a coin

Tree diagram to represent the sample space of four successive tosses of a coin

H

H

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T T

T

H

H H

H

T

T T

T

H

H

H

T T

T

H

H H

H

T

T T

T

T

H

HHHH

HHHT

HHTH

HHTT

HTHH

HTHT

HTTH

HTTT

THHH

THHT

THTH

THTT

TTHH

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TTTT

There are 3 chips in a bag, two red and one blue. You shake the bag and pull out one without looking. It could be red or blue. Then you pull out another one. What could the colours of the first and the second chip be? Work out the possible combinations and answer the question. What is most likely to happen? It is most likely that I

would pull out 2 red chipsIt is most likely that I would pull out 1 red and 1 blue chipBoth of these are equally likely

Lecoutre, 1996

There are 3 chips in a bag, two red and one blue. You shake the bag and pull out one without looking. It could be red or blue. Then you pull out another one. What could the colours of the first and the second chip be? Work out the possible combinations and answer the question. What is most likely to happen? It is most likely that I

would pull out 2 red chipsIt is most likely that I would pull out 1 red and 1 blue chipBoth of these are equally likely

1st pulled out

R

R

B

2nd pulled out

Lecoutre, 1996

There are 6 possible outcomes.

One R and one W is twice as probable as two Rs

But only around 50% of various groups of undergraduates gave the correct answer

Lecoutre, 1996

Difficulty in working out all the possible outcomes

R1-W R2 -W R1 -R2 R2 -R1 W -R1 W -R2

Other problems that could be removed by working on the sample space

Kahneman & Tversky’s representativeness error (BGBGGB more likely than BBBGGG) is actually a failure in inspecting the sample space or in wrongly aggregating the sample space.

Van Dooren et al. demonstrated that most adolescents judge as correct the statement I roll a die 12 times. My chance of getting at least two 6s in these 12 throws is 3times as great as my chance of getting at least two 6s if I roll the die 4 times. They would probably not make this “linear” error if they worked out the sample space for 4 and 12 throws, since this would show that there is no linear relationship between the number of throws and the possibility of two sixes.

Aggregating

Counting out the number of possible alternatives is often not enough.

It is often necessary to form these alternatives into categories

This causes a lot of difficulties, particularly when the basic alternatives are equiprobable but category membership is not.

Abrahamson’s scoop task

Abrahamson gave 12-year old children big box of green and blue balls and a 4-ball scoop

And then asked them about the probability of the outcomes of a scoop

His questions were not about single outcomes but about categories of outcomes e.g. how likely is it to be 3G and 1B?

So he was asking them to aggregate

The sample space: 16 possible scoops

Abrahamson’s results This is a very difficult task for 12-year olds. Their commonest mistake is to say that the 5 aggregated categories

are equiprobable. The children’s justifications seem to show that the cause of their

difficulty is having to deal with two levels of data – the 16 individual equiprobable outcomes and the 5 non-equiprobable categories – at the same time and yet keep them separate.

We need much more research (on adults as well as children) on this possibility

A confusion between levels of aggregation probably also causes the ‘representativeness’ mistake

Back to the future

The topic of sample space raises a cognitive question , which is “how systematically can children think about the future”?

There is no data at all on children’s ability to list all the possible events in a particular context.

There is research on children’s counterfactual reasoning about alternative possibilities (what would have happened if Napoleon hadn’t had indigestion at Waterloo?) but this is post-hoc and about deterministic chains of events

3. The quantification of probability

The difference between extensive and intensive quantities

Some quantities, like weight and volume, are extensive: if you add one to another the quantity increases: if I have a kilo of apples and put another kilo into it, the basket is now heavier

Other quantities are intensive: they do not obey the same rules of addition: if two pieces of wood have the same density, and I join them together, the new object has the same density as its two parts

Intensive quantities are based on proportions Probability is an intensive quantity: the proportion of a

particular event to all the possible events

Box A contains 3 marbles of which 1 is white and 2 are black.

Box B contains 7 marbles of which 2 are white and 5 black.

You have to draw a marble from one of the boxes with your eyes covered. From which box should you draw if you want a white marble?”

PISA, 2003

The calculation is a proportional one: the proportion of white in A is.33: the proportion of white in B is .29Only 27% of a large group of German 15-year olds got the right answer: worse than chance level

Comparing probabilities

The cards are shuffled several times and then put into the box where they belong.

Tick the box where there’s a better chance of picking a circle, or tick the It doesn’t matter which box if you think that you’ve the same chance of picking a circle in one box as in the other

.

box 1 box 2

These are the cards in box 1

These are the cards in box 2

.It doesn’t matter which box

or

Tick box 1 or box 2 or tick the It doesn’t matter which box

or

.box 1 box 2

It doesn’t matter which box

These are the cards in box 1

These are the cards in box 2

Tick box 1 or box 2 or tick the It doesn’t matter which box

or

box 1 box 2It doesn’t matter which box

These are the cards in box 1

These are the cards in box 2

Tick box 1 or box 2 or tick the It doesn’t matter which box

box 1 box 2

It doesn’t matter which box

These are the cards in box 1

These are the cards in box 2

Problems which did not need a proportional solution

An example is the trial in which there were the same number of circles (6 in both) in both boxes, but an unequal number of squares (5 in one and 6 in the other)

In this comparison, the child could solve the problem just by directly comparing the number of squares in the two sets

Piaget &

Problems which did need a proportional solution

In other trials the number of circles and the number of squares both differed between the two boxes. This made the problem a genuinely proportional one.

These were much the harder of the two kinds of problem in our pre-test.

Piaget & Inhelder report that the childen who did solve such problems reached their solution by calculating ratios, not fractions.

This chimes with our work on other intensive quantities

General conclusions

Children’s understanding of and interest in fairness seem a good start for working on their learning about randomness

The importance of analysing the sample space first has been badly underestimated in past research. We urgently need research on teaching children how to do this

Their preference for ratios over fractions gives us an important lead into how to teach them to quantify probabilities.

The relationship between these three kinds of knowledge needs investigation: this should combine intervention studies with longitudinal research

Bravado 

 

Have I not walked without an upward look

Of caution under stars that very well

Might not have missed me when they shot and fell?

It was a risk I had to take – and took.

 

Robert Frost