U
n
i
t
7
–
P
ro
babili
ty
&
S
t
a
t
i
st
ic
s
—
F
u
ll De
pt
h S
tu
d
y
G
u
ide
(
NC
Ma
t
h
2
CDM
)
1
.
PROBABILITY
—
Dee
p
Di
v
e
Sa
mp
le S
p
ace
s
The
s
a
mp
le
sp
ace i
s
t
he
s
e
t
o
f
ALL
poss
ible
out
c
om
e
s
o
f a
n
e
xp
e
r
i
m
e
nt
.
E
x
a
mp
le
:
ro
lli
n
g a
die
→
s
a
mp
le
sp
ace
=
{
1
,
2
,
3
,
4
,
5
,
6
}
E
x
a
mp
le
:
fl
i
pp
i
n
g
2
c
o
i
ns
→
s
a
mp
le
sp
ace
=
{
HH
,
HT
,
TH
,
TT
}
Li
st
i
n
g
t
he
s
a
mp
le
sp
ace ca
r
ef
u
ll
y
(
a li
st
,
t
able
,
or
tr
ee diag
r
a
m
)
i
s
o
f
t
e
n
t
he ke
y
first
st
e
p
—
m
i
s
c
ount
i
n
g
out
c
om
e
s
i
s
t
he
#
1
sour
ce
o
f e
rrors
i
n
pro
babili
ty
pro
ble
ms
.
The
or
e
t
ical P
ro
babili
ty
Ba
s
ed
on
t
he
s
a
mp
le
sp
ace
—
w
ha
t
SHOULD ha
pp
e
n
m
a
t
he
m
a
t
icall
y
,
a
ssum
i
n
g all
out
c
om
e
s
a
r
e e
qu
all
y
likel
y
.
F
ormu
la
:
P
(
e
v
e
nt
)
=
(#
o
f fa
vor
able
out
c
om
e
s
)
/
(#
o
f
tot
al
poss
ible
out
c
om
e
s
)
E
x
a
mp
le
:
P
(
ro
lli
n
g a
n
e
v
e
n
num
be
r
on
a die
)
=
3
/
6
=
1
/
2
E
xp
e
r
i
m
e
nt
al P
ro
babili
ty
Ba
s
ed
on
ACTUAL
tr
ial
s
/
da
t
a c
o
llec
t
ed
—
w
ha
t
DID ha
pp
e
n
.
F
ormu
la
:
P
(
e
v
e
nt
)
=
(#
o
f
t
i
m
e
s
e
v
e
nt
o
cc
urr
ed
)
/
(
tot
al
#
o
f
tr
ial
s
)
E
x
a
mp
le
:
A c
o
i
n
i
s
fl
i
pp
ed
50
t
i
m
e
s
,
la
n
di
n
g head
s
28
t
i
m
e
s
→
e
xp
e
r
i
m
e
nt
al P
(
head
s
)
=
28
/
50
=
0
.
56
Ke
y
di
st
i
n
c
t
i
on
:
t
he
or
e
t
ical i
s
calc
u
la
t
ed i
n
ad
v
a
n
ce f
rom
r
ea
son
i
n
g
;
e
xp
e
r
i
m
e
nt
al c
om
e
s
f
rom
r
eal
(
or
s
i
mu
la
t
ed
)
da
t
a
.
A
s
t
he
num
be
r
o
f
tr
ial
s
i
n
c
r
ea
s
e
s
,
e
xp
e
r
i
m
e
nt
al
pro
babili
ty
t
e
n
d
s
to
ge
t
cl
os
e
r
to
t
he
or
e
t
ical
pro
babili
ty
(
La
w
o
f La
r
ge N
um
be
rs
).
C
omp
le
m
e
nt
R
u
le
The c
omp
le
m
e
nt
o
f a
n
e
v
e
nt
A
(
wr
i
tt
e
n
A
'
or
"
not
A
")
i
s
e
v
e
ryt
hi
n
g i
n
t
he
s
a
mp
le
sp
ace
t
ha
t
ISN
'
T
A
.
E
v
e
ry
out
c
om
e i
s
ei
t
he
r
i
n
A
or
i
n
i
ts
c
omp
le
m
e
nt
—
n
e
v
e
r
b
ot
h
,
n
e
v
e
r
n
ei
t
he
r
—
so
t
he
two
pro
babili
t
ie
s
al
w
a
ys
add
up
to
1
.
F
ormu
la
:
P
(
A
)
+
P
(
A
')
=
1
,
or
r
ea
rr
a
n
ged
:
P
(
A
')
=
1
−
P
(
A
)
Whe
n
to
us
e i
t
:
a
ny
t
i
m
e
you
'
r
e a
s
ked f
or
"
a
t
lea
st
on
e
,"
"
non
e
,"
or
"
not
,"
i
t
'
s
usu
all
y
mu
ch
fa
st
e
r
to
fin
d
t
he c
omp
le
m
e
nt
a
n
d
su
b
tr
ac
t
f
rom
1
t
ha
n
to
add
up
e
v
e
ry
i
n
di
v
id
u
al ca
s
e
.
E
x
a
mp
le
:
P
(
ro
lli
n
g a die a
n
d ge
tt
i
n
g a
num
be
r
le
ss
t
ha
n
6
)
=
1
−
P
(
ro
lli
n
g a
6
)
=
1
−
1
/
6
=
5
/
6
E
x
a
mp
le
("
a
t
lea
st
on
e
"
—
t
he cla
ss
ic c
omp
le
m
e
nt
tr
igge
r
):
A c
o
i
n
i
s
fl
i
pp
ed
3
t
i
m
e
s
.
Fi
n
d
P
(
a
t
lea
st
on
e head
s
).
C
omp
le
m
e
nt
=
P
(
no
head
s
a
t
all
)
=
P
(
TTT
)
=
(
1
/
2
)
³
=
1
/
8
P
(
a
t
lea
st
on
e
head
s
)
=
1
−
1
/
8
=
7
/
8
2
.
COMPOUND PROBABILITY
—
Dee
p
Di
v
e
C
ompoun
d e
v
e
nts
i
nvo
l
v
e
two
or
mor
e
out
c
om
e
s
c
om
bi
n
ed
("
or
"
/
"
a
n
d
").
Addi
t
i
on
R
u
le
(
f
or
"
OR
")
U
s
ed
w
he
n
fin
di
n
g
t
he
pro
babili
ty
t
ha
t
EITHER
o
f
two
e
v
e
nts
ha
pp
e
ns
.
N
on
-
mutu
all
y
e
x
cl
us
i
v
e e
v
e
nts
(
ca
n
ha
pp
e
n
a
t
t
he
s
a
m
e
t
i
m
e
—
t
he
y
ov
e
r
la
p
):
P
(
A
or
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A a
n
d B
)
(
su
b
tr
ac
t
t
he
ov
e
r
la
p
so
i
t
i
sn
'
t
d
ou
ble
-
c
ount
ed
)
M
utu
all
y
e
x
cl
us
i
v
e e
v
e
nts
(
ca
nnot
ha
pp
e
n
a
t
t
he
s
a
m
e
t
i
m
e
—
no
ov
e
r
la
p
):
P
(
A
or
B
)
=
P
(
A
)
+
P
(
B
)
E
x
a
mp
le
:
d
r
a
w
i
n
g a ca
r
d
t
ha
t
i
s
a Ki
n
g OR a Hea
rt
(
ov
e
r
la
p
=
Ki
n
g
o
f Hea
rts
):
P
(
Ki
n
g
or
Hea
rt
)
=
4
/
52
+
13
/
52
−
1
/
52
=
16
/
52
=
4
/
13
M
u
l
t
i
p
lica
t
i
on
R
u
le
(
f
or
"
AND
")
U
s
ed
w
he
n
fin
di
n
g
t
he
pro
babili
ty
t
ha
t
BOTH
o
f
two
e
v
e
nts
ha
pp
e
n
.
I
n
de
p
e
n
de
nt
e
v
e
nts
(
on
e d
o
e
sn
'
t
a
ff
ec
t
t
he
ot
he
r
):
P
(
A a
n
d B
)
=
P
(
A
)
×
P
(
B
)
De
p
e
n
de
nt
e
v
e
nts
(
on
e DOES a
ff
ec
t
t
he
ot
he
r
—
e
.
g
.
not
r
e
p
laci
n
g a
n
i
t
e
m
):
P
(
A a
n
d B
)
=
P
(
A
)
×
P
(
B gi
v
e
n
A al
r
ead
y
ha
pp
e
n
ed
)
E
x
a
mp
le
(
i
n
de
p
e
n
de
nt
):
P
(
fl
i
pp
i
n
g head
s
AND
ro
lli
n
g a
6
)
=
1
/
2
×
1
/
6
=
1
/
12
E
x
a
mp
le
(
de
p
e
n
de
nt
,
no
r
e
p
lace
m
e
nt
):
d
r
a
w
i
n
g
2
ace
s
i
n
a
row
f
rom
a deck
w
i
t
h
out
r
e
p
lace
m
e
nt
:
P
=
(
4
/
52
)
×
(
3
/
51
)
—
t
he
s
ec
on
d f
r
ac
t
i
on
cha
n
ge
s
beca
us
e
on
e ace a
n
d
on
e
ca
r
d a
r
e al
r
ead
y
g
on
e
.
Wi
t
h Re
p
lace
m
e
nt
vs
.
Wi
t
h
out
Re
p
lace
m
e
nt
Thi
s
i
s
t
he
s
i
n
gle bigge
st
cl
u
e f
or
decidi
n
g i
n
de
p
e
n
de
nt
vs
.
de
p
e
n
de
nt
w
he
n
a
pro
ble
m
i
nvo
l
v
e
s
d
r
a
w
i
n
g i
t
e
ms
mu
l
t
i
p
le
t
i
m
e
s
.
WITH
r
e
p
lace
m
e
nt
:
t
he i
t
e
m
i
s
put
back bef
or
e
t
he
n
e
xt
d
r
a
w
.
The
tot
al c
ount
r
e
s
e
ts
each
t
i
m
e
,
so
e
v
e
ry
d
r
a
w
ha
s
t
he SAME
pro
babili
t
ie
s
—
t
he e
v
e
nts
a
r
e INDEPENDENT
.
F
ormu
la
p
a
tt
e
rn
:
P
(
A a
n
d B
)
=
P
(
A
)
×
P
(
B
)
—
s
a
m
e de
nom
i
n
a
tor
b
ot
h
t
i
m
e
s
.
E
x
a
mp
le
:
d
r
a
w
a ca
r
d
,
r
e
p
lace i
t
,
d
r
a
w
agai
n
.
P
(
two
Ki
n
g
s
)
=
(
4
/
52
)
×
(
4
/
52
)
WITHOUT
r
e
p
lace
m
e
nt
:
t
he i
t
e
m
i
s
ke
pt
out
.
The
tot
al c
ount
s
h
r
i
n
k
s
b
y
1
(
a
n
d
t
he c
ount
o
f
w
ha
t
e
v
e
r
w
a
s
d
r
a
wn
s
h
r
i
n
k
s
too
,
if d
r
a
w
i
n
g
t
he
s
a
m
e
typ
e agai
n
)
—
t
he e
v
e
nts
a
r
e
DEPENDENT
.
F
ormu
la
p
a
tt
e
rn
:
P
(
A a
n
d B
)
=
P
(
A
)
×
P
(
B
|
A
)
—
t
he
s
ec
on
d f
r
ac
t
i
on
'
s
de
nom
i
n
a
tor
(
a
n
d
som
e
t
i
m
e
s
num
e
r
a
tor
)
cha
n
ge
s
.
E
x
a
mp
le
:
d
r
a
w
a ca
r
d
,
d
o
NOT
r
e
p
lace
i
t
,
d
r
a
w
agai
n
.
P
(
two
Ki
n
g
s
)
=
(
4
/
52
)
×
(
3
/
51
)
Q
u
ick
r
ec
o
g
n
i
t
i
on
t
able
:
Ph
r
a
s
e i
n
pro
ble
m
Re
p
lace
m
e
nt
?
I
n
de
p
e
n
de
nt
or
de
p
e
n
de
nt
?
"
w
i
t
h
r
e
p
lace
m
e
nt
"
Ye
s
I
n
de
p
e
n
de
nt
"
r
e
p
lace
s
i
t
"
/
"
puts
i
t
back
"
Ye
s
I
n
de
p
e
n
de
nt
"
w
i
t
h
out
r
e
p
lace
m
e
nt
"
N
o
De
p
e
n
de
nt
"
d
o
e
s
not
r
e
p
lace
"
/
"
kee
ps
i
t
out
"
N
o
De
p
e
n
de
nt
Se
p
a
r
a
t
e
/
unr
ela
t
ed e
v
e
nts
(
c
o
i
n
+
die
,
sp
i
nn
e
r
+
sp
i
nn
e
r
)
N
/
A I
n
de
p
e
n
de
nt
T
wo
d
r
a
ws
f
rom
t
he
s
a
m
e g
roup
,
no
m
e
nt
i
on
o
f
r
e
p
laci
n
g
A
ssum
e N
o
De
p
e
n
de
nt
3
.
CONDITIONAL PROBABILITY
—
Dee
p
Di
v
e
C
on
di
t
i
on
al
pro
babili
ty
=
t
he
pro
babili
ty
o
f a
n
e
v
e
nt
GIVEN
t
ha
t
a
not
he
r
e
v
e
nt
ha
s
al
r
ead
y
o
cc
urr
ed
.
W
r
i
tt
e
n
P
(
B
|
A
)
—
"
pro
babili
ty
o
f B gi
v
e
n
A
."
T
wo
-
Wa
y
Table
s
A
t
able
t
ha
t
or
ga
n
i
z
e
s
da
t
a b
y
two
ca
t
eg
or
ie
s
a
t
on
ce
(
rows
a
n
d c
o
l
umns
),
w
i
t
h
row
/
c
o
l
umn
tot
al
s
.
Fi
n
di
n
g c
on
di
t
i
on
al
pro
babili
ty
f
rom
a
two
-
w
a
y
t
able
:
P
(
B
|
A
)
=
(
v
al
u
e i
n
t
he
sp
eci
fi
c cell
)
/
(
tot
al f
or
row
or
c
o
l
umn
A
—
w
hiche
v
e
r
c
on
di
t
i
on
i
s
"
gi
v
e
n
")
Ke
y
habi
t
:
t
he
wor
d
"
gi
v
e
n
"
t
ell
s
you
w
hich
tot
al
to
di
v
ide b
y
—
r
e
str
ic
t
your
de
nom
i
n
a
tor
to
ONLY
t
he
row
or
c
o
l
umn
m
a
t
chi
n
g
t
he c
on
di
t
i
on
,
not
t
he f
u
ll g
r
a
n
d
tot
al
.
E
x
a
mp
le
t
able
(
O
wns
a
p
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.
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s
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y
a
r
d
):
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