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Dominion» Forums » General

Subject: Math and graphs time! Here's how many games it takes to see every Dominion card rss

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Mike Hulsebus
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This started out because with the new second edition and a copy of Dominion Empires & Adventures coming my way, I was curious how long it would take for me to me actually see all the new cards. I made a quick excel spreadsheet that tried to run some simulations and it seemed like if I played 100 games, I would end up with about 1 card that I still hadn't seen play, but I didn't have anything precise.

MATH TO THE RESCUE
My brother knows his stuff when it comes to math, and he was kind enough to run some numbers. As it turns out, this problem is very similar to The Coupon Collector's Problem, which you can learn more about here https://en.wikipedia.org/wiki/Coupon_collector%27s_problem

For my question (how long does it take to see 68 new cards) the answer about 124 games on average.

But let's say you have never played Dominion and bought all the Dominion that currently exists. How many games would you need to play in order to see all the cards once? Here's a graph of 50,000 simulations

Mean: 165.0975 games
Median: 159 games
Lowest: 90 games
Highest: 490 games


With how much people talk about how many combinations of cards there are, I thought this was another interesting bit of information to add to the discussion. I have 324 logged plays of Dominion at the time of this posting, and looks like I have quite a few plays ahead of me until I've seen everything once the new cards arrive.

I want to take a moment to talk about a few assumptions that the data above makes
1) It ignores cards like Young Witch that go and get another card
2) It assumes that every time you play, you reshuffle the randomizer deck to make it completely random
3) It assumes every time you play, you just go and get 10 random cards (instead of some of the optional rules that some people use like rolling a die to see whether or not you will use potion cards this game)
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Chad Weaver
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Not sure how you are getting to 271 cards. After Empires, there were 260. With the second editions and Sauna/Avanto, there are either 275 or 263, depending how you feel about the replaced first edition cards.
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Chris Fell
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Cool I am glad you shared this with us.
Now if you could be so kind to ask your brother to run the sims a few more times to see what the numbers look like for just the original Dominion sets (no Empires & Adventures).
 
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Shawn Garbett
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Like the plot. It's got some interesting properties. I'd never heard of the coupon collector's problem. I spent an hour or so trying to solve the analytical form of the distribution. Actually, I can solve it for smaller problems, the numbers in this get too large and I was trying some math tricks to get it tamed. I can already tell I'm going to be obsessed with this for a bit. Especially given the shape of that curve, it might explain the cell division checkpoint better than using the EMG model I proposed a few years back.

The closed form expectation of average is know for this problem. It should be n*H_n / c, where n is the number of total cards, H_n is the nth Harmonic number and c is the number of cards per draw.


> 271*sum(1/(1:271)) / 10
[1] 167.5099


Which is quite close to your simulated average. I bet if you bootstrapped your estimate, the 95% C.I. will contain the above value.
 
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Mike Hulsebus
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Chad, you may be right: Looks like 274 may be the right number?

Base set 25
Revised 7
Intrigue 25
Revised 7
Seaside 26
Alchemy 12
Prosperity 25
Cornucopia 13
Hinterlands 26
Dark Ages 35
Guilds 13
Empires 24
Adventures 30
English Promos 6
 
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Alex G

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Coupon collecting is neat. Actually, with the exception that the "equal probability" is almost completely wrong, it's a useful way to model some things in software testing (see http://www.evosuite.org/wp-content/papercite-data/pdf/icst12... for an example where the equal probability problem isn't so bad), and part of why AI methods aren't as useful in test generation as you might think. AI/machine learning is often tuned to the "win chess!" or "win Go!" type goal, while testing is more like "win Go in every interestingly different possible way!"
 
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BorderCon
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BorderCon was a blast this year!!!
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Hmm...I like turtles!
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Matthew
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mikehulsebus wrote:
Chad, you may be right: Looks like 274 may be the right number?

Base set 25
Revised 7
Intrigue 25
Revised 7
Seaside 26
Alchemy 12
Prosperity 25
Cornucopia 13
Hinterlands 26
Dark Ages 35
Guilds 13
Empires 24
Adventures 30
English Promos 6

It's 275. You're forgetting the new promo, Sauna/Avanto.
 
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Henry Swantner
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If you'd like some real fun, calculate the number of unique sets of 10 kingdom cards, unique games, that can be played. Though it's just a calculation without any probability or methodology involved.
We did that quite a few expansions ago and it would already take to the heat death of the universe to play them all.
 
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Jason Hunt
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RevanFan wrote:
mikehulsebus wrote:
Chad, you may be right: Looks like 274 may be the right number?

Base set 25
Revised 7
Intrigue 25
Revised 7
Seaside 26
Alchemy 12
Prosperity 25
Cornucopia 13
Hinterlands 26
Dark Ages 35
Guilds 13
Empires 24
Adventures 30
English Promos 6

It's 275. You're forgetting the new promo, Sauna/Avanto.


They did specifically call out "English" Promos. Sauna/Avanto is not yet available in English.
 
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Salvador C. Majoral
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Argentona, Barcelona
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jivjov wrote:
Sauna/Avanto is not yet available in English.


I don't think it is even known whether the bottom ones will be called avanto or ice hole (or something else), is it?
 
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Matt E
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salva wrote:
jivjov wrote:
Sauna/Avanto is not yet available in English.


I don't think it is even known whether the bottom ones will be called avanto or ice hole (or something else), is it?

Yes, and it's "Avanto".
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Ron Laufer
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mikehulsebus wrote:
2) It assumes that every time you play, you reshuffle the randomizer deck to make it completely random
That's why we don't do that. We put the 10 we just played backwards at the back and don't reshuffle until we've gone through all of them. So it will take us 28 games, once we catch up and get Empires and the upgrade packs.
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