Understanding all outcomes from the throw of to die is vital to many games.
The table below lists each outcome from two dice throws. Not all outcomes are unique. For example, there are two ways to get a one
on one dice and a two on the other dice.
The one could be on the first dice and the two on the second dice or the
order could be reversed.
Questions:
·
List all unique outcomes from the two dice roll
experiments.
 · What is the probability of each outcome?
 · What is the probability of getting snake eyes? (Snake eyes is 1 on each dice.) How does the probability of getting snake eyes compare to the probability of getting (1,2).
 · There are six outcomes where the first dice is equal to the second dice. What is the probability the throw of the two dice results in two identical numbers?
 · There are 15 outcomes where the two dice throws result in two different values. What is the probability that the values of the two dice throws differ in value?
Outcomes from two throw
of the Dice


First Dice

Second Dice

Probability of Each
Outcome


1

1

1

0.0278

2

2

1

0.0278

3

3

1

0.0278

4

4

1

0.0278

5

5

1

0.0278

6

6

1

0.0278

7

1

2

0.0278

8

2

2

0.0278

9

3

2

0.0278

10

4

2

0.0278

11

5

2

0.0278

12

6

2

0.0278

13

1

3

0.0278

14

2

3

0.0278

15

3

3

0.0278

16

4

3

0.0278

17

5

3

0.0278

18

6

3

0.0278

19

1

4

0.0278

20

2

4

0.0278

21

3

4

0.0278

22

4

4

0.0278

23

5

4

0.0278

24

6

4

0.0278

25

1

5

0.0278

26

2

5

0.0278

27

3

5

0.0278

28

4

5

0.0278

29

5

5

0.0278

30

6

5

0.0278

31

1

6

0.0278

32

2

6

0.0278

33

3

6

0.0278

34

4

6

0.0278

35

5

6

0.0278

36

6

6

0.0278

Answer: In most games like Risk or Monopoly it does
not matter if one throws two dice and gets (i,j) or (j,i). For instance in monopoly if you throw (1,2)
or (2,1) you move three. Similarly,
both of these throws in Risk give the player a max value of 2 and a min value
of 1.
Below I have listed all unique outcomes from two throws of
the dice and the probability of each outcome.
Unique Outcomes From Two
Dice Throws Regardless of Dice Outcomes


Outcome #

Unique Pair

Probability


1

1

1

0.0278

2

1

2

0.0556

3

1

3

0.0556

4

1

4

0.0556

5

1

5

0.0556

6

1

6

0.0556

7

2

2

0.0278

8

2

3

0.0556

9

2

4

0.0556

10

2

5

0.0556

11

2

6

0.0556

12

3

3

0.0278

13

3

4

0.0556

14

3

5

0.0556

15

3

6

0.0556

16

4

4

0.0278

17

4

5

0.0556

18

4

6

0.0556

19

5

5

0.0278

20

5

6

0.0556

21

6

6

0.0278

1.0000

Note the probability of getting snake eyes (1,1) is 1/32
while the probability of getting unique pair (1,2) is 1/18. This is because there is only one way to get
snake eyes but two ways to get (1,2).
(The only way to get snake eyes is to get 1 on both throws. The two ways to get (1,2) are one on the
first dice and two on the second dice and 2 on the first dice and 1 on the
second dice.)
There are six unique outcomes when the dice are equal  (1,1), (2,2), (3,3), (4,4), (5,5) and
(6,6). Each of these outcomes has a
probability of 1/36. The probability
that the two die throws results in identical outcomes is 6/36 or 1/6.
There are 15 unique outcomes when the two dice have
different values. (See the table above
for each of these outcomes.) The
likelihood of each outcome is 1/18. The
likelihood that one of the 15 outcomes where the dice values differs occurs is
15/18 or 5/6.
The probability that dice values are equal is 1/6. The complement of equal dice values is the
event dice values differ. The
probability of the complement is 5/6.
In the next post I will use what we have learned here about
the probability of two dice throws to figure out the likelihood of different
outcomes in the game of Risk when the attacking army attacks with two armies
and the defending army defends with two armies.
Note in a previous post I looked at what happens in the game
of Risk when two attacking armies are greeted by one defending army.
Previous Problem in the Game of Risk:
Authors Note: I wrote a short book Statistical Applications
of Baseball in 1997. It was well reviewed
by the journal Chance. A number of
people have told me they have benefited from this book. The book is very inexpensive.
Statistical Applications of Baseball
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