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BinomialDistribution
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Mathematics
Outcomes from Dice
Example Notebook
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Simulate dice tosses for various combinations of dice
Simple Cases
(
8
)
Here are possible values for a 6-sided die:
I
n
[
1
]
:
=
T
a
b
l
e
[
i
,
{
i
,
6
}
]
O
u
t
[
1
]
=
{
1
,
2
,
3
,
4
,
5
,
6
}
These are possible totals for 2 6-sided dice:
I
n
[
2
]
:
=
F
l
a
t
t
e
n
[
T
a
b
l
e
[
i
+
j
,
{
i
,
6
}
,
{
j
,
6
}
]
]
O
u
t
[
2
]
=
{
2
,
3
,
4
,
5
,
6
,
7
,
3
,
4
,
5
,
6
,
7
,
8
,
4
,
5
,
6
,
7
,
8
,
9
,
5
,
6
,
7
,
8
,
9
,
1
0
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
}
Here's a histogram of the total values:
I
n
[
3
]
:
=
H
i
s
t
o
g
r
a
m
[
%
,
{
1
}
]
O
u
t
[
3
]
=
Results for 3 dice:
I
n
[
4
]
:
=
F
l
a
t
t
e
n
[
T
a
b
l
e
[
i
+
j
+
k
,
{
i
,
6
}
,
{
j
,
6
}
,
{
k
,
6
}
]
]
O
u
t
[
4
]
=
{
3
,
4
,
5
,
6
,
7
,
8
,
4
,
5
,
6
,
7
,
8
,
9
,
5
,
6
,
7
,
8
,
9
,
1
0
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
4
,
5
,
6
,
7
,
8
,
9
,
5
,
6
,
7
,
8
,
9
,
1
0
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
5
,
6
,
7
,
8
,
9
,
1
0
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
}
Histogram of values for 3 dice:
I
n
[
5
]
:
=
H
i
s
t
o
g
r
a
m
[
%
,
{
1
}
]
O
u
t
[
5
]
=
Results for a pair of 12-sided (dodecahedral) dice:
I
n
[
6
]
:
=
F
l
a
t
t
e
n
[
T
a
b
l
e
[
i
+
j
,
{
i
,
1
2
}
,
{
j
,
1
2
}
]
]
O
u
t
[
6
]
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
4
,
5
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
5
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
6
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
7
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
8
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
9
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
2
0
,
1
0
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
2
0
,
2
1
,
1
1
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
2
0
,
2
1
,
2
2
,
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
2
0
,
2
1
,
2
2
,
2
3
,
1
3
,
1
4
,
1
5
,
1
6
,
1
7
,
1
8
,
1
9
,
2
0
,
2
1
,
2
2
,
2
3
,
2
4
}
Histogram of the results:
I
n
[
7
]
:
=
H
i
s
t
o
g
r
a
m
[
%
,
{
1
}
]
O
u
t
[
7
]
=
Histogram of values for 3 12-sided dice:
I
n
[
8
]
:
=
H
i
s
t
o
g
r
a
m
[
F
l
a
t
t
e
n
[
T
a
b
l
e
[
i
+
j
+
k
,
{
i
,
1
2
}
,
{
j
,
1
2
}
,
{
k
,
1
2
}
]
]
,
{
1
}
]
O
u
t
[
8
]
=
T
h
e
M
o
r
e
G
e
n
e
r
a
l
C
a
s
e
(
9
)
B
e
y
o
n
d
C
o
m
p
u
t
i
n
g
T
o
t
a
l
s
(
5
)
See Also
CoinTossSimulator
IdealCoinTossStatistics
Related Symbols
Tuples
BinomialDistribution
Publisher Information
Contributed by:
Wolfram Staff
