Posted by: thebylog | May 1, 2012

Risk Probabilities

A couple weeks ago, I was talking to my father and he mentioned that he had recently played a game of Risk. A natural question arose: What is the probability of succeeding when you attack with three dice against two defending dice? I thought long enough about an analytical solution to deduce that it would not be very easy (maybe I am wrong?), so I simulated it. The results:

  • You will lose two armies 25.66% of the time
  • You will lose one army 32.75% of the time
  • You will lose zero armies 41.59% of the time

Because you need to know.

[Of course, you can find this sort of thing on the web. Here, for instance, is a table that suggests that if you have five or more armies, you have a better than 50% chance of winning, as long as your opponent has no more armies than you.]



  1. May I extrapolate? Thus, the game of Risk rewards the aggressive use of force, even against equal forces. Defense is not effective unless you can stockpile resources, which the game makes difficult unless you conquer and hold significant territory. Preemptive strikes are statistically better poised to win the day than careful defense.

    That said, I liked the game. There might have been reasons.

    I think the neocons may have grown up playing Risk.

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