Tuesday, March 22, 2016

Probability Problems in the Game of Risk -- Question One

 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 1-0.4167 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 (1-p) 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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