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 attackingnation 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.
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