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 uncertainty
Identifying 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 predicted Older children predicted
progressive mixing
continued order
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?
15 purple marbles
15 yellow marbles
A purple marble
A yellow marble
Each colour is just as likely
Difficulty in understanding the independence of
random events
 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 answer
Chiesi & Primi
Percentages for the different answers in Chiesi
& Primi’s task
Positive
recency
Negative
recency
8yrs
Correct
answer
0
10yrs
40
College
student
41
Percentages for the different answers in Chiesi
& Primi’s task
Positive
recency
Negative
recency
Correct
answer
8yrs
66
34
0
10yrs
30
30
40
College 16
student
43
41
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
The sample space of four successive tosses of
a coin
HHHT
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HHTT
HHTH
HTHT
HTHH
HTTT
HTTH
TTTT
TTTH
TTHT
TTHH
THTT
THTH
THHT
THHH
Tree diagram to represent the sample space of
four successive tosses of a coin
HHHH
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T
H
H
HHHT
HHTH
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T
T
HHTT
H
T
H
H
T
HTHH
HTHT
HTTH
T
H
T
HTTT
THHH
H
THHT
THTH
T
H
H
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T
THTT
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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
chips
It is most likely that I
would pull out 1 red and
1 blue chip
Both 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?
1st pulled
out
R
R
B
2nd pulled
out
It is most likely that I
would pull out 2 red
chips
It is most likely that I
would pull out 1 red and
1 blue chip
Both of these are equally
likely
Lecoutre, 1996
Difficulty in working out all the possible
outcomes
 R1-W
 R2 -W
 R1 -R2
 R2 -R1
 W -R1
 W -R2
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
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
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
Comparing probabilities
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?”
The calculation is a proportional one:
the proportion of white in A is.33:
the proportion of white in B is .29
Only 27% of a large group of German 15year olds got the right answer: worse than
PISA, 2003
chance level
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
These are the
cards in box 1
or
These are the
cards in box 2
.
box 1
box 2
It doesn’t matter
. which box
Tick box 1 or box 2 or tick the
It doesn’t matter which box
These are the
cards in box 1
box 1
or
These are the
cards in box 2
box 2
It doesn’t matter
. which box
Tick box 1 or box 2 or tick the
It doesn’t matter which box
These are the
cards in box 2
These are the
cards in box 1
or
box 1
It doesn’t matter which box
box 2
Tick box 1 or box 2 or tick the
It doesn’t matter which box
These are the
cards in box 1
box 1
These are the
cards in box 2
box 2
It doesn’t matter which box
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