RISK: Can Strategy Affect Probability of Winning?

[Astudious]

My friend proposed to me this problem, as we were playing the boardgame RISK (rules written here http://www.hasbro.com/common/instruct/risk.pdf if you don't know em, I guess the relevant bit is really p10-11).

Let's say I want to capture 2 territories, A and B, with 2 and 3 defending pieces on them respectively. I start off with 6 pieces in a country neighbouring both A and B. So, it's up to me my strategy: to capture A (2), then B (3), or the reverse. Will it make a difference (however small) to my probability of winning (succeeding in capturing both), which way I go?

Gut instinct looking at the boardgame is "no", but instinct looking at the maths is "yes". On the other hand I don't have a clue how to calculate such probabilities quantitatively and I'm not expecting it (even calculating the outcome of a single battle seems tricky http://www4.stat.ncsu.edu/~jaosborn/research/RISK.pdf though we could probably use the results if we needed).

I just want to know, if someone can suggest (and to some degree explain) from mathematical instinct, whether there will be a difference and if so which is the ideal path?

 

[mfb]

You can use the table on page 5 to write down the full relevant game tree with probabilities for the first battle. It is not too large. Page 6 then gives the probabilities for the second battle.

As different army sizes give different probabilities to win/lose individual rounds, I would expect a difference, but I don't know in which direction.