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Probability and the game of RISK

Date: 04/21/2000 at 19:48:46
From: Josh Peretti
Subject: Probability and the game of RISK.

In the classic game of Risk the attacker attacks with 3 dice and the 
defender defends with 2 dice. What is the probability that the 
defender will lose 2 battalions? What is the probability that they 
will each lose 1 battalion? What is the probability that the attacker 
will lose 2 battalions?

It's a very well-known game, but just in case you don't know, here are 
the relevant rules: The attacker first rolls his 3 dice and discards 
the lowest number. The defender then rolls his 2 dice. At this point 
the attacker matches his highest number with the highest number of the 
defender and his lowest number with the defender's lowest number. If 
the attacker's highest number is greater than the defender's highest 
number, the attacker wins. If it is equal or lesser, the attacker 
loses. The same is done for the pair of lowest numbers.

I've been discussing this with a group of junior-high math teachers, 
and we have no idea how to figure it out.

Josh Peretti

Date: 04/22/2000 at 17:49:15
From: Doctor Mitteldorf
Subject: Re: Probability and the game of RISK.

Dear Josh -

One way to go about this is the brute force technique. I encourage you 
to actually do this, because it is straightforward and transparent, 
and it will help you understand where any more abstract approach is 
coming from.

The brute force technique would be to model the situation with a 
computer. You'll have 5 different dice, each with values from 1 to 6, 
so you can program with 5 nested loops

     for a1:=1 to 6 do
        for a2:= 1 to 6 do
           for a3:=1 to 6 do
              for b1:=1 to 6 do
                 for b2:= 1 to 6 do

Inside all these loops, you'll have some if...then statements about 
what happens. First, compare to find the higher and lower of the two 
b's; then find the highest and middle of the two a's. Compare a_high 
and b_high, a_mid and b_low, and tally up the results. This tally will 
be run for each of the 7776 (6*6*6*6*6) possible outcomes. The program 
will count how many of these 7776 possibilities result in victory by 
the attacker, etc. The program will take half an hour of thinking in 
which you will come to understand the situation you're modeling much 
more intimately. It will run in less than the blink of an eye.

A more abstract approach would be to come up with probability 
distributions for the attacker's dice and the defender's dice. Let's 
start with the defender's high die. There are 6*6 = 36 possible 
outcomes to consider. In 11 of them, the highest die is a 6, so the 
probability of the higher of the defender's dice being a 6 is 11/36. 
How did I count this? The first die can be 6 and the second die can be 
anything (that's 6 possibilities) or the first die can be 1 ... 5 and 
the second die can be a 6 (5 more possibilities.) Similarly, you can 
tabulate the probabilities for the defender's highest die:

     Higher     cases
      die     out of 36
       6         11
       5          9
       4          7
       3          5
       2          3 
       1          1

The probabilities for the lower of the 2 dice are just reversed; the 
probability of a 1 is 11/36, of a 2 is 9/36, etc. Can you see why?

I'll leave it to you to do the similar but more complicated analysis 
for the attacker. Start with the number of cases out of 6*6*6 = 216 in 
which the highest of the three dice is a 6. There are 91 of them, 
hence the probability for the highest die being a 6 is 91/216. 
Continue to find the complete probability table for the highest die. 
Then move to the middle die, and compute its probability table. Of 
course, you'll discard the lowest die.

Now you have a probability table for the highest of three and for the 
higher of 2. From that, you can pair each of the 36 pairings, add them 
up, and have the result for the higher die; similarly for the lower 
die. It's a good deal of tabulation and calculation no matter how you 
do it, but it's a shortcut compared to what the computer does, which 
is to take each of the 7776 cases separately.

The results I get are as follows:

     Attacker wins: 2890/7776 (0.372) 
     Defender wins: 2275/7776 (0.293) 
     A and D Split: 2611/7776 (0.336)

Thanks for the review of the Risk rules. The last time I played Risk 
was about 1965, I think. I hope you'll continue and complete the 
calculation, both as a computer program and as a hand calculation. 
Maybe the two will even come out the same. In any case, I hope you'll 
write back and either report your completed results, or tell me 
exactly what you did and how far you got with it before you got stuck.

- Doctor Mitteldorf, The Math Forum   
Associated Topics:
High School Probability

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