## Tuesday, March 22, 2016

### Probability in the Game of Risk – Problem Two

Probability in the Game of Risk – Problem Two

This second risk probability problem looks at outcomes when the attacking country attacks with two armies and the defending country fights with one army.   The reader might want to review combat resolution rules in Risk.   I placed a few examples of risk combat resolution rules at this post.

Readers who want more on the rules of Risk can go to this web site among other sites.

Question Two:  List the probability distribution function describing outcomes from the game of risk when an attacking nation attacks with two armies and the defending nation defends with one army?

What is the expected number of victories for the attacking nation when this experiment is repeated 10 times?

What is the expected number of victories for the defending nation when this experiment is repeated 10 times?

What are the variances of attacking nation and the variance of defending nation victories in this situation?

Answer:  There are two possible outcomes when the attacking nation attacks with two armies and the defending nation attacks with one army.

If either of the attacking nation dice is greater than or equal to the defending nation dice the attacking nation will win one army.

If both attacking nation dice are less than or equal to the defending nation dice the defending nation wins one army.

The defending nation cannot win or lose more than one army because it is only committing one army to the battle.

There are 216 combinations of attacking nation and defending nation armed conflict outcomes for this situation.   There are 36 (6 x 6) combinations of attacking nation dice rolls when the attacking nation uses 2 armies.   Each of these attacking nations combinations is matched with 6 defending nation outcomes so total combinations is 216 (6 x 36.)

For each combination, I compare the best (highest) attacking nation dice roll to the defending nation dice roll.   If the best attacking-nation dice roll is larger than the defending nation dice roll the defending nation wins.

The results from the 216 possible dice rolls are summarized below.

 Risk outcomes when 2 attacking armies meet one defending army Attacking  Nation Wins 125 Defending Nation Wins 91 Probability Attacking Nation Wins 0.579 Probability Defending Nation Wins 0.421

The expected number of wins from 10 trials for the attacking nation is 5.79.

The expected number of wins from 10 trials for the defending nation is 4.21.

The variance of wins for 10 trials is 10 x 0.579 x 0.421, or 2.438.   The variance of wins is the same for attacking and defending nations.

Authors Note:  I will be traveling for the next week.   When I return I will pick up my work on the probability behind the game of risk.

My book Solving Financial Problems in Excel is potentially useful for two reasons.  First, the problems are interesting.   Second, I show how to use Excel finance functions.  Please try this book if you need or want this information.

Also, remember to look at the first risk probability problem.