Monday, October 14, 2013

A Unique and Special Snowflake

As I was making dinner, I got to thinking about Aristocratic games and, in particular, the notion that PCs are supposed to be unique and special.  Being a math nerd, it got me wondering, "What is the probability that, rolling on 3d6 in order, you will roll a set of stats that you have rolled before?  How many ACKS PCs do you need to roll (in expectation) before you will get one which is not unique-from-your-perspective in terms of ability scores?"

This, it turns out, is not immediately obvious to me.  It looks like a variation on the coupon-collector's problem, except that the coupons have varying base probability, which is not really something accounted for in the typical formulation of the problem.

This is going to be fun!  But I should perhaps leave solving it for another night, ideally one where I do not need to work the next morning.

(And if someone finds a known general form, don't tell me.  Well, maybe tell me that one exists, but no spoilers.  And if I solve this one, I may move on to 'functionally unique', in terms of ability score modifier bands.  Actually that might be a simpler form to solve first...)

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