15
Teaching & Learning Plans Plan 2: Probability and Relative Frequency Junior Certificate Syllabus Leaving Certificate Syllabus
Teaching & Learning Plans
Plan 2: Probability and Relative Frequency
Junior Certificate Syllabus Leaving Certificate Syllabus
The Teaching & Learning Plans are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic.
Learning Outcomes outline what a student will be able to do, know and understand having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the topic.
Lesson Interaction is set out under four sub-headings:
i. Student Learning Tasks – Teacher Input: This section focuses on teacher input and gives details of the key student tasks and teacher questions which move the lesson forward.
ii. Student Activities – Possible and Expected Responses: Gives details of possible student reactions and responses and possible misconceptions students may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning.
iv. Checking Understanding: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
© Project Maths Development Team 2009 www.projectmaths.ie 1
Teaching & Learning Plan 2: Probability and Relative Frequency
AimsTo introduce the concept of ‘outcomes’ of an ‘event’•
To estimate likelihood of occurrence of events•
To establish the ‘Sample Space’ as the set of all possible outcomes•
Prior Knowledge Students should have prior knowledge (from T and L Plan 1 and/or from primary school) of some terms associated with chance and uncertainty. They should be familiar with probability expressed as a fraction or decimal in the range 0 to 1, or as a percentage in the range 0% to 100%.
Learning OutcomesAs a result of studying this topic, students will be able to
understand and use the following terminology: trial, outcome, set of •all possible outcomes, relative frequency, event, theoretical probability, equally likely outcomes
list all the possible outcomes when rolling a fair die•
recognise that the outcomes on successive throws of a die are independent •of each other
calculate, from experimental results, the relative frequency for each •outcome and note how it approaches the theoretical probability as the number of trials increases
understand the concept of a fair die•
distinguish equally likely outcomes from those which are not•
associate probability with long-run relative frequency•
Teaching & Learning Plan 2: Probability and Relative Frequency
© Project Maths Development Team 2009 www.projectmaths.ie 2
Relationship to Junior Certificate SyllabusSub-topics Ordinary Level Higher Level (Inc.OL)1.5 Counting Listing outcomes of
experiments in a systematic way
1.6 Concepts of probability
Recognise that probability is a measure on a scale of 0-1 (and 0-100%) of how likely an event is to occur.
Estimate probabilities from experimental data.
Associate the probability of an event with its long-run, relative frequency.
1.7 Outcomes of simple random processes
Apply the principle that, in the case of equally likely outcomes, the probability is given by the number of outcomes of interest divided by the total number of outcomes.
Relationship to Leaving Certificate Syllabus
Sub-topics Foundation Level Ordinary Level1.1 Counting List outcomes of an
experiment.
Apply the fundamental principle of counting.
1.2 Concepts of probability
Recognise that probability is a measure on a scale of 0-1 of how likely an event is to occur.
Estimate probabilities from experimental data.
Associate the probability of an event with its long-run, relative frequency.
1.3 Outcomes of random processes
Apply the principle that, in the case of equally likely outcomes, the probability is given by the number of outcomes of interest divided by the total number of outcomes.
Resources RequiredA die for each pair of students
Teaching & Learning Plan 2: Probability and Relative Frequency
© Project Maths Development Team 2009 www.projectmaths.ie 3
Introducing the Topic
An “Event”
Following from our last lesson we know that, directly or indirectly, probability or chance plays a role in a wide range of activities. We often make statements which involve terms such as the likelihood or the chance of occurrence of an event – what do we mean by an ‘event’?
‘It will probably rain today’. The event is ‘It will rain today.’•
‘Though we are sending the national team to the Olympics, we cannot •confidently predict that we shall win a gold medal’. The event is ‘We shall win a gold medal’.
‘There is a chance that Roy Keane will manage the Irish Football team.’ The •event is ‘Roy Keane will manage the Irish football team.
Each statement above suggests an event whose occurrence or non-occurrence involves an element of uncertainty.
Estimating the chance of an ‘event’ occurring?
Because of past information or currently available statistics for an event, we can predict, with some degree of confidence, what the outcome of the event will be.
The past performance of Brazil’s football team in the World Cup can help us to estimate the probability of the team winning the next World Cup (they have won 5 out of the last 12 world cups: 1958, 1962, 1970, 1994 and 2002).
Thus, for example, we may make the statement (a) above if most of the days we have observed recently were rainy days.
Associating numbers with phrases like “very likely” and “probably”
In conversation we might say that it was ‘very warm’ yesterday. Would this have the same meaning for a person living in central Australia as for someone living in Birr? How can we be clearer about what “very warm” means? The expert from the Meteorological Office would state the maximum temperature in degrees C, thus quantifying the situation.
Similarly a person might describe himself as ‘having big feet’, but when it comes to buying shoes, a more exact description, i.e. shoe size, is needed.
Terms like ‘most likely’ and ‘probably’ are too vague for many purposes; so, ways of measuring probability have been devised.
Teaching & Learning Plan 2: Probability and Relative Frequency
© Project Maths Development Team 2009 www.projectmaths.ie 4
Real Life ContextApart from gambling, (mention Las Vegas) the theory of probability can help us make relatively reasonable choices in our daily lives.
Students choose college courses which have a high probability of leading to employment after college.
Many people have left New Orleans for good because there is a high probability of devastating hurricanes in that region.
People don’t smoke because there is a high probability of their developing cancer as a result of smoking.
Probability has widespread use in business, science and industry. Its uses range from the determination of life insurance premiums to the description of the behaviour of molecules in a gas and also the prediction of the outcomes of an election.
Teac
hing
& L
earn
ing
Plan
2: P
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ve F
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ency
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5
Less
on
In
tera
ctio
nSt
uden
t Le
arni
ng T
asks
: Te
ache
r In
put
Stud
ent A
ctiv
itie
s: P
ossi
ble
and
Expe
cted
Res
pons
esTe
ache
r’s S
uppo
rt a
nd
Act
ions
Chec
king
Und
erst
andi
ng
Wh
en y
ou
to
ss a
die
wh
at
»ar
e th
e p
oss
ible
ou
tco
mes
?1,
2, 3
, 4, 5
, 6•
Ask
an
ind
ivid
ual
stu
den
t.
»Te
ll th
e cl
ass
that
we
call
this
set
of
all p
oss
ible
o
utc
om
es “
the
sam
ple
sp
ace”
.
Wh
en y
ou
to
ss a
die
wh
ich
»
nu
mb
er is
mo
st li
kely
to
ap
pea
r?
Wh
ich
is le
ast
likel
y?
»
Wri
te d
ow
n y
ou
r an
swer
»
ind
ivid
ual
ly o
n t
he
top
of
Stu
den
t A
ctiv
ity
1A.
Wo
rkin
g in
pai
rs, r
oll
a d
ie
»30
tim
es (
i.e. 3
0 tr
ials
) an
d
fill
in c
olu
mn
s 2
and
3 o
n
Stu
den
t A
ctiv
ity
1B.
Mo
st li
kely
____
____
__
•
Leas
t lik
ely_
____
____
_•
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
1. By
wal
kin
g a
rou
nd
en
sure
»
that
eve
ryo
ne
has
mad
e a
gu
ess.
G
ive
a d
ie t
o e
ach
pai
r. »
Do
th
e st
ud
ents
’ an
swer
s »
sho
w m
isco
nce
pti
on
s?
This
may
infl
uen
ce y
ou
r d
ecis
ion
on
th
e n
ext
par
t –
wh
eth
er t
o a
sk s
tud
ents
to
d
o 3
0 o
r 50
ro
lls o
f th
e d
ie.
Do
th
e st
ud
ents
kn
ow
ho
w
»to
do
tal
ly m
arks
?
Are
stu
den
ts u
sin
g c
orr
ect
»te
rmin
olo
gy,
e.g
. wh
at w
as
the
ou
tco
me
for
that
tri
al?
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
6
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Fill
you
r re
sult
s o
nto
th
e »
Mas
ter
Tab
le A
on
th
e b
oar
d w
hen
yo
u h
ave
com
ple
ted
co
lum
ns
2 an
d
3.
As
stu
den
ts fi
nis
h t
hey
fill
»re
sult
s in
to t
he
Mas
ter
Tab
le A
on
th
e b
oar
d i.
e.
the
freq
uen
cies
of
1’s,
2’s
, et
c. f
or
each
gro
up
.
Put
an A
3 si
ze c
op
y o
f »
Mas
ter
Tab
le A
(as
in
Stu
den
t A
ctiv
ity
2A)
on
th
e b
oar
d.
Co
llect
all
dic
e. W
hile
»
colle
ctin
g d
ice
no
te w
hat
th
e la
st c
olu
mn
ad
ds
up
to
fo
r ea
ch g
rou
p.
Loo
k at
th
e »
Mas
ter
Tab
le
A, d
id e
very
gro
up
get
th
e sa
me
nu
mb
er o
f 1’
s o
r 2’
s et
c. f
or
30 t
rial
s?
Is t
her
e a
tren
d a
pp
eari
ng
? »
Stu
den
ts c
an s
ee t
hat
th
eir
»re
sult
s ar
e n
ot
the
sam
e b
ut
sim
ilar.
Stu
den
ts m
igh
t su
gg
est
»ca
lcu
lati
ng
th
e av
erag
e n
um
ber
of
1’s
etc.
Stu
den
ts c
an c
alcu
late
th
e »
aver
age
nu
mb
ers
of
1’s
etc.
an
d in
form
th
e cl
ass.
Giv
e g
rou
ps
a m
inu
te t
o
»lo
ok
at t
he
resu
lts
fro
m t
he
clas
s an
d c
om
par
e th
e cl
ass
resu
lts
to t
hei
r o
wn
.
Do
stu
den
ts n
oti
ce a
tre
nd
»
in t
he
ou
tco
mes
?
Cal
cula
te t
he
rela
tive
»
freq
uen
cy f
or
each
o
utc
om
e (c
olu
mn
4)
and
th
e co
rres
po
nd
ing
p
erce
nta
ges
in c
olu
mn
5
(Stu
den
t A
ctiv
ity
1B)
Stu
den
ts c
alcu
late
th
e »
rela
tive
fre
qu
enci
es a
nd
sh
ou
ld s
ee t
hat
th
ey a
re a
ll al
mo
st e
qu
al.
Cir
cula
te c
hec
kin
g
»st
ud
ents
’ cal
cula
tio
ns
aski
ng
qu
esti
on
s w
her
e n
eces
sary
.
Are
stu
den
ts b
egin
nin
g t
o
»se
e th
at a
ll o
utc
om
es a
re
equ
ally
like
ly?
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
7
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
If y
ou
ad
d u
p a
ll th
e va
lues
»
in t
he
last
2 c
olu
mn
s in
yo
ur
ow
n t
able
(St
ud
ent
Act
ivit
y 1B
) w
hat
do
th
ey
add
up
to
?
Fill
thes
e an
swer
s in
to
»St
ud
ent
Act
ivit
y 1C
.
Ther
e is
a c
on
nec
tio
n
»b
etw
een
pro
bab
ility
an
d
rela
tive
fre
qu
ency
wh
ich
w
e w
ill c
om
e b
ack
to la
ter
in t
he
less
on
.
1 an
d 1
00%
•G
ive
stu
den
ts a
few
»
mo
men
ts t
o c
on
sid
er t
hei
r an
swer
s an
d t
hen
ask
d
iffe
ren
t g
rou
ps
for
thei
r an
swer
s, s
o t
hat
th
e cl
ass
real
ises
eve
ryo
ne
go
t 1.
Ask
th
ose
wh
o d
idn
’t g
et
»1
to r
ech
eck
thei
r an
swer
s.
Cir
cula
te a
nd
hel
p w
ith
q
uer
ies.
Hav
e al
l stu
den
ts r
ealis
ed
»th
at t
he
sum
of
all t
he
pro
bab
iliti
es f
or
a sa
mp
le
spac
e is
1?
Fro
m t
he
last
less
on
, »
wh
at d
id 1
mea
n o
n t
he
pro
bab
ility
sca
le?
Wh
at o
utc
om
es a
re y
ou
»
cert
ain
to
get
wit
h t
his
die
?
“1”
mea
nt
cert
ain
ty.
•
1 o
r 2
or
3 o
r 4
or
5 o
r 6.
•
The
pro
bab
ility
of
get
tin
g
on
e o
f th
ese
nu
mb
ers
is 1
.
Wh
ile w
alki
ng
aro
un
d
»ch
eck
that
eve
ryo
ne
has
ad
ded
up
last
2 c
olu
mn
s an
d h
as g
ot
1 an
d 1
00%
.
Wh
at d
o y
ou
no
tice
»
abo
ut
you
r re
sult
s? D
o
they
co
nfi
rm /r
efu
te y
ou
r p
red
icti
on
?
Wri
te d
ow
n a
co
ncl
usi
on
»
(Stu
den
t A
ctiv
ity
1D)
and
ref
er b
ack
to y
ou
r p
red
icti
on
an
d w
hy
you
m
ade
that
pre
dic
tio
n.
Stu
den
ts w
rite
up
th
eir
»co
ncl
usi
on
s.
Dif
fere
nt
gro
up
s re
ad o
ut
»th
eir
con
clu
sio
n a
nd
if it
is
in a
gre
emen
t w
ith
th
eir
pre
dic
tio
n, g
ive
reas
on
s w
hy
they
mad
e th
at
pre
dic
tio
n.
Cir
cula
te g
ivin
g a
dvi
ce
»w
her
e n
eces
sary
, ask
ing
so
me
gro
up
s to
sta
te
thei
r co
ncl
usi
on
s to
th
e cl
ass
wh
ile c
hec
kin
g t
hat
ea
ch g
rou
p h
as a
wri
tten
co
ncl
usi
on
.
Was
eac
h g
rou
p a
ble
to
»
mak
e a
wri
tten
co
ncl
usi
on
?
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
8
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Ho
w m
igh
t yo
u m
od
ify
»th
is e
xper
imen
t so
th
at
you
co
uld
hav
e m
ore
co
nfi
den
ce in
yo
ur
con
clu
sio
n?
Stu
den
ts m
ay s
ug
ges
t m
ore
»
tria
ls.
Ho
w m
igh
t w
e lo
ok
at t
he
»re
sult
s fo
r a
larg
er n
um
ber
o
f tr
ials
?
Stu
den
ts s
ug
ges
t u
sin
g t
he
»se
t o
f cl
ass
resu
lts
fro
m:
Mas
ter
Tab
le A
on
th
e b
oar
d.
Do
stu
den
ts e
xpec
t le
ss
»va
riat
ion
an
d c
lear
er t
ren
ds
wit
h m
ore
tri
als?
Co
py
»M
aste
r Ta
ble
A f
rom
th
e b
oar
d a
nd
fill
in c
lass
re
sult
s.
Cal
cula
te t
he
rela
tive
»
freq
uen
cy f
or
each
o
utc
om
e (c
olu
mn
4)
and
th
e co
rres
po
nd
ing
p
erce
nta
ges
(co
lum
n 5
) fo
r ea
ch o
utc
om
e.
No
w t
hat
yo
u h
ave
ove
r »
300
tria
ls c
an y
ou
see
an
y p
atte
rn e
mer
gin
g?
Wri
te a
co
ncl
usi
on
, ref
erri
ng
ag
ain
to
yo
ur
pre
dic
tio
n.
Stu
den
ts m
ake
ou
t th
eir
»o
wn
Mas
ter
Tab
le a
nd
do
th
e ca
lcu
lati
on
s (S
tud
ent
Act
ivit
y 2B
) an
d w
rite
a
con
clu
sio
n (
Stu
den
t A
ctiv
ity
2C)
com
par
ing
it
wit
h t
hei
r co
ncl
usi
on
fro
m
30 t
rial
s.
This
tim
e d
iffe
ren
t st
ud
ents
»
read
ou
t th
eir
con
clu
sio
ns
no
tin
g a
ny
dif
fere
nce
s w
ith
th
e la
st t
ime
– tr
end
cle
arer
w
ith
mo
re t
rial
s.
Stu
den
ts n
ote
th
at t
he
»re
lati
ve f
req
uen
cy o
f ea
ch
ou
tco
me
is t
he
sam
e i.e
. 1/
6. a
pp
rox.
Wal
k ar
ou
nd
an
d c
hec
k »
the
pre
dic
tio
n a
gai
nst
th
e ex
per
imen
tal r
esu
lts
and
as
k st
ud
ents
wh
y th
ey
mad
e th
at p
red
icti
on
.
Ask
th
ose
wh
o t
ho
ug
ht
6 »
was
th
e h
ard
est
to g
et a
nd
w
hy
they
th
ou
gh
t th
is.
(Exp
erie
nce
from
boa
rd g
ames
, bi
gges
t nu
mbe
r?)
Did
stu
den
ts s
ee a
cle
arer
»
pat
tern
of
alm
ost
eq
ual
ly
likel
y o
utc
om
es f
rom
th
e la
rger
nu
mb
er o
f tr
ials
?
Did
stu
den
ts w
ho
had
»
tho
ug
ht
that
6 w
as h
ard
er
to g
et a
ckn
ow
led
ge
thei
r m
isco
nce
pti
on
?
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
9
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Ind
ivid
ual
ly d
raw
a g
rap
h
»o
n g
rap
h p
aper
usi
ng
th
e d
ata
fro
m t
he
Mas
ter
Tab
le.
Wri
te a
co
ncl
usi
on
. »
Stu
den
ts c
ho
ose
th
e ty
pe
»o
f g
rap
h t
o d
raw
, fo
r ex
amp
le a
bar
ch
art.
Stu
den
ts m
ay d
raw
»
dif
fere
nt
gra
ph
s.
Stu
den
ts c
om
men
t o
n t
he
effe
ctiv
enes
s o
f th
eir
gra
ph
in
illu
stra
tin
g a
tre
nd
. St
ud
ents
co
uld
als
o d
raw
a
gra
ph
of
thei
r o
wn
res
ult
s co
mp
arin
g it
to
th
e cl
ass
Mas
ter
Tab
le r
esu
lts.
Co
ncl
usi
on
s fr
om
gra
ph
s »
sho
uld
be
con
sist
ent
wit
h
tho
se f
rom
tab
les.
Cir
cula
te g
ivin
g h
ints
/hel
p
»w
her
e n
eces
sary
.
Cir
cula
te r
ead
ing
th
e »
con
clu
sio
ns,
ask
ing
a f
ew
stu
den
ts t
o r
ead
ou
t th
eirs
to
th
e cl
ass.
No
te: P
ossi
ble
use
of A
utog
raph
–s
imul
atin
g th
row
ing
a di
e th
ousa
nds
of t
imes
.
Bec
ause
all
ou
tco
mes
are
»
equ
ally
like
ly c
an y
ou
th
ink
of
a w
ord
to
des
crib
e a
die
lik
e th
is if
it w
ere
use
d in
a
gam
e w
hic
h g
ave
ever
yon
e th
e sa
me
chan
ce?
Som
e st
ud
ents
may
co
me
•u
p w
ith
th
e w
ord
“fa
ir”.
G
ive
the
wo
rd ‘u
nb
iase
d’,
if
»th
e st
ud
ents
do
n’t
su
gg
est
it a
nd
rei
tera
te t
hat
wh
at is
m
ean
t b
y a
fair
die
is t
hat
al
l ou
tco
mes
are
eq
ual
ly
likel
y.
The
men
u c
ho
ices
fo
r »
tod
ay’s
lun
ch a
re p
izza
, ro
ast
bee
f an
d s
alad
. Is
the
pro
bab
ility
th
at I
will
ch
oo
se p
izza
= 1
/3?
Yes
•
I am
alle
rgic
to
ch
eese
so
•
the
pro
bab
ility
th
at I
wo
uld
ch
oo
se p
izza
is z
ero
.
I am
a v
eget
aria
n a
nd
•
wo
uld
nev
er c
ho
ose
bee
f.
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
10
Stud
ent
Lear
ning
Tas
ks: T
each
er In
put
Stud
ent A
ctiv
itie
s: P
ossi
ble
and
Expe
cted
Res
pons
esTe
ache
r’s S
uppo
rt a
nd
Act
ions
Chec
king
U
nder
stan
ding
Wh
at is
dif
fere
nt
abo
ut
this
an
d t
he
»o
utc
om
es f
or
the
fair
die
?
Ho
w m
igh
t w
e fi
nd
ou
t th
e »
pro
bab
ility
?
They
are
no
t al
l eq
ual
ly
•lik
ely.
We
mig
ht
do
a s
urv
ey b
ut
•co
uld
on
ly g
et e
stim
ated
va
lues
of
the
pro
bab
ility
as
ou
tco
mes
are
no
t eq
ual
ly li
kely
.
Giv
e st
ud
ents
a m
inu
te
»to
dis
cuss
th
is in
th
eir
gro
up
an
d t
hen
ask
d
iffe
ren
t g
rou
ps.
If
som
eon
e an
swer
s ‘y
es’,
ask
if a
nyo
ne
else
has
a
sug
ges
tio
n.
Can
stu
den
ts
»d
isti
ng
uis
h b
etw
een
‘e
qu
ally
like
ly ‘a
nd
‘n
ot
equ
ally
like
ly’?
Can
yo
u t
hin
k o
f o
ther
exa
mp
les
of
»si
tuat
ion
s w
her
e th
e o
utc
om
es a
re
no
t eq
ual
ly li
kely
?
In h
ors
e ra
cin
g –
dep
end
s •
on
fo
rm, j
ock
ey, e
tc.
5 p
eop
le g
oin
g f
or
an
•in
terv
iew
do
no
t al
l h
ave
an e
qu
al c
han
ce o
f g
etti
ng
th
e jo
b.
Load
ed d
ie ,
die
wit
h, s
ay,
•tw
o 1
’s a
nd
tw
o 2
’s, e
tc.
Pose
th
e q
ues
tio
n t
o
»th
e cl
ass
and
tak
e an
swer
s as
stu
den
ts p
ut
up
th
eir
han
ds.
Did
stu
den
ts a
ctiv
ely
»p
arti
cip
ate
in t
he
dis
cuss
ion
, giv
ing
lots
o
f id
eas
and
sh
ow
ing
u
nd
erst
and
ing
?
The
last
ap
pro
ach
is k
no
wn
as
»th
e ‘e
xper
imen
tal ‘
or
‘em
pir
ical
ap
pro
ach
’ to
cal
cula
tin
g p
rob
abili
ties
. H
ow
ever
, we
did
no
t ca
lcu
late
p
rob
abili
ties
bu
t in
stea
d c
alcu
late
d
the
rela
tive
fre
qu
enci
es –
so
wh
at is
th
e co
nn
ecti
on
wit
h p
rob
abili
ty?
Has
an
yon
e g
ot
any
idea
s o
n t
his
? N
ote
: (St
uden
ts w
ho m
ay s
ee t
he
conn
ectio
n ne
ed t
o be
allo
wed
say
so)
.
Wh
at d
o y
ou
th
ink
wo
uld
be
the
»va
lue
of
the
rela
tive
fre
qu
enci
es if
we
did
mo
re a
nd
mo
re t
rial
s?
Exp
erim
ents
giv
e •
esti
mat
es o
f th
eore
tica
l p
rob
abili
ty b
ased
on
th
e re
lati
ve f
req
uen
cy o
f ea
ch o
utc
om
e. R
elat
ive
freq
uen
cy t
end
s to
war
ds
the
pro
bab
ility
as
the
nu
mb
er o
f tr
ials
get
s ve
ry
larg
e.
They
wo
uld
get
clo
ser
and
•
clo
ser
to 1
/6 a
nd
hen
ce
wo
uld
all
be
the
sam
e.
Teac
hing
& L
earn
ing
Plan
2: P
roba
bilit
y an
d Re
lati
ve F
requ
ency
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
11
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
Act
ions
Chec
king
U
nder
stan
ding
Bas
ed o
n w
hat
yo
u n
ow
»
kno
w a
bo
ut
pro
bab
ility
an
d it
s re
lati
on
ship
to
lon
g
term
rel
ativ
e fr
equ
ency
, an
d a
lso
bas
ed o
n t
he
bar
ch
art
rep
rese
nti
ng
th
e fr
equ
ency
of
each
o
utc
om
e, c
an y
ou
no
w fi
ll in
th
e la
st c
olu
mn
6 in
th
e M
aste
r Ta
ble
A?
Stu
den
ts fi
ll in
1/6
fo
r th
e »
pro
bab
ility
of
each
ou
tco
me.
If t
he
nu
mb
er o
f tr
ials
has
»
no
t b
een
su
ffici
entl
y la
rge
stu
den
ts m
ay n
ot
see
that
th
e o
utc
om
es a
re e
qu
ally
lik
ely
so e
ith
er r
epea
t w
ith
a
larg
er n
um
ber
of
tria
ls o
r u
se o
f a
com
pu
ter
gen
erat
ed
sim
ula
tio
n. T
his
sh
ou
ld
con
vin
ce s
tud
ents
th
at a
ll o
utc
om
es a
re e
qu
ally
like
ly.
Hav
e al
l stu
den
ts
»b
een
ab
le t
o
com
ple
te t
he
pro
bab
ility
co
lum
n
corr
ectl
y?
Refl
ecti
on
Wri
te d
ow
n 3
item
s yo
u
»le
arn
ed a
bo
ut
pro
bab
ility
to
day
. W
rite
do
wn
an
yth
ing
yo
u
»fo
un
d d
iffi
cult
.W
rite
do
wn
an
y q
ues
tio
n
»yo
u m
ay h
ave.
All
ou
tco
mes
are
eq
ual
ly li
kely
1.
w
hen
to
ssin
g a
fai
r d
ie.
Rel
ativ
e fr
equ
ency
= f
req
uen
cy
2.
of
ou
tco
me
div
ided
by
the
nu
mb
er o
f tr
ials
.
Equ
ally
like
ly o
utc
om
es a
ll 3.
h
ave
an e
qu
al p
rob
abili
ty o
f o
ccu
rrin
g.
Rel
ativ
e fr
equ
ency
=
4.
pro
bab
ility
fo
r an
infi
nit
e n
um
ber
of
tria
ls w
her
e o
utc
om
es a
re e
qu
ally
like
ly.
Cir
cula
te a
nd
tak
e n
ote
•
par
ticu
larl
y o
f an
y q
ues
tio
ns
stu
den
ts h
ave
and
hel
p t
hem
to
an
swer
th
ese.
Teaching & Learning Plan 2: Probability and Relative Frequency
© Project Maths Development Team 2009 www.projectmaths.ie 12
Student Activity 1
Number Which Appears on Die
(Outcome of Trial)
How Many Times Did This Happen?
Use Tally Marks to Help You Count
Total (Frequency)
Fraction of Total Scores (Relative
Frequency)
Frequency no. of trials
Percentages of Total Scores
123456
Totals
Student Activity 1D Conclusion: (Refer to prediction)
Student Activity 1B
Student Activity 1A My prediction:
Which number is most likely to appear________________?
Which number is least likely to appear________________?
Student Activity 1C
The sum of all the relative frequencies is ________________?
The sum of all the percentages is ________________?
x100 Frequency no. of trials
Teaching & Learning Plan 2: Probability and Relative Frequency
© Project Maths Development Team 2009 www.projectmaths.ie 13
Student Activity 2Student Activity 2A Master Table
Student Activity 2C Conclusion: (Refer to prediction)
Student Activity 2B
The sum of all the relative frequencies is ________________?
The sum of all the percentages is ________________?
Number Which
Appears on Die
(Outcome of Trial)
How Many Times Did
This Happen? (Frequency) Each Class
Group Writes in its Result
Total of Frequencies
Fraction of Total Scores (Relative
Frequency)
Percentage of Total Scores
Probability of Each
Outcome
1 E.g. 5+6+5+....
2
3
4
5
6
x100 Frequency no. of trials total
300 Frequency sample size=
