I am planning several
probability problems on the game of risk.
This is the first one.
Readers who want to play with these problems need to know
the rules of the game. There are
several web sites that explain the game.
I obtained combat resolution rules from one site and placed
in a post below.
Readers who want more on the rules can go to this web site
among other sites.
Question One: The attacking army attacks with one army and
the defending army defends with one army.
What is the probability the attacking army wins the contest and the
probability the defending army wins the contest? What are the expected number of wins and
variance of wins by attacking army and defending army if this one versus one
combat occurs 10 times?
Answer to question
one: The attacker throws one dice. The defender throws on dice. The winner has the larger die outcome but
ties go to the defender.
The Bernoulli or binary distribution models the outcome of
one war.
There are a total of 36 outcomes from the two dice
throws. The outcomes are listed
below.
The attacking army wins 15 of the outcomes.
The defending army wins 21 of the outcomes.
The probability that the attacking army wins the war is 0.4167.
The probability the defending army wins the war is 10.4167
or 0.5833.
If this experiment is repeated 10 time we have a binomial
distribution with n=10 and p as calculated above.
Expected wins for the attacking and defending army are 10 x
0.4167 and 10 x 0.5833 or 4.167 and 5.833 respectively.
The variance of wins is n x p x (1p) or 10 x 0.4167 x
0.5833, which is 2.43.
One attacking army and
one defending army


Combination
Number

Defending
Country

Attacking
Country

Attacking
Country Wins

1

1

1

0

2

1

2

1

3

1

3

1

4

1

4

1

5

1

5

1

6

1

6

1

7

2

1

0

8

2

2

0

9

2

3

1

10

2

4

1

11

2

5

1

12

2

6

1

13

3

1

0

14

3

2

0

15

3

3

0

16

3

4

1

17

3

5

1

18

3

6

1

19

4

1

0

20

4

2

0

21

4

3

0

22

4

4

0

23

4

5

1

24

4

6

1

25

5

1

0

26

5

2

0

27

5

3

0

28

5

4

0

29

5

5

0

30

5

6

1

31

6

1

0

32

6

2

0

33

6

3

0

34

6

4

0

35

6

5

0

36

6

6

0

Sum

15


Count

36


Prob Attacking Country
Wins

0.4167


Prod Defending Country
Wins

0.5833

Authors Note: Learning how to solve financial problems in
Excel is an important skill. My Kindle
book on this topic is a useful resource.
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