Dominion Strategy Forum
Dominion => Dominion General Discussion => Topic started by: GendoIkari on June 28, 2016, 08:01:59 pm
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How do you determine the percentage of kingdoms that will have at least 1 of a given set of cards? Rough estimate is fine; ignoring things like Young Witch and Black Market.
Specifically, I'm curious what percentage of kingdoms will allow you to end the game with debt (not counting the situation of buying a debt-cost card for an immediate 3-pile).
Aside from forcing your opponent to do it via Possession, you would need to have a gainer in the Kingdom. So, aside from the task of listing out the gainers, I don't know the formula for determining the percentage of Kingdoms.
This comes from a discussion over here (https://boardgamegeek.com/thread/1595867/why-no-penalty-end-game-debt)...
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You work out the chance of a kingdom having none of those cards and then subtract that from 1.
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Facetious as it sounds, DG is pretty much spot on.
If there are m = 260 Kingdom cards in total, and you choose n = 10 of them to form a Kingdom, then the probability that none of k are selected is C(m - k, n) / C(m, n), and so the probability that at least one is selected is 1 - C(m - k, n) / C(m, n).
For example, there are 3 Gathering cards, so the chance that none at least one is included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Unfortunately, that calculation is a bit harder if you also want to consider Events and Landmarks, and even moreso if you want to look at intersections of two categories. At that point, you're possibly better off programming a really simplistic Kingdom selector and running it a million times to get a Monte Carlo estimate.
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For example, there are 3 Gathering cards, so the chance that none are included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Wait, that's not right...? The chance for none of a 3 card set to be in any given game has to be way higher than that. How often do you have games that don't include any of Kings Court, Goons, and Grand Market?
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For example, there are 3 Gathering cards, so the chance that none are included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Wait, that's not right...? The chance for none of a 3 card set to be in any given game has to be way higher than that. How often do you have games that don't include any of Kings Court, Goons, and Grand Market?
He mixed up the explanation of the results. The ~11.1% is the chance that at least one of the cards in the set of 3 is in the Kingdom. My gut tells me that probability is close to reality in my total sample space of games.
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He mixed up the explanation of the results. The ~11.1% is the chance that at least one of the cards in the set of 3 is in the Kingdom. My gut tells me that probability is close to reality in my total sample space of games.
That sounds completely believable for the probability of actually having any of the 3.
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For example, there are 3 Gathering cards, so the chance that none are included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Wait, that's not right...? The chance for none of a 3 card set to be in any given game has to be way higher than that. How often do you have games that don't include any of Kings Court, Goons, and Grand Market?
About 89% of the time, probably, if I understand the point of the calculations.
So if you choose 3 random cards, one of them should be in about 11,1% games and be absent kn 89%.
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I want to know what the probability is of seeing any Event/Landmark.
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I want to know what the probability is of seeing any Event/Landmark.
Limiting to 2 total events/landmarks, the chance of any particular event/landmark is 2.7%.
Without limiting to 2, I get:
Chance of 0 Events/Landmarks: 14.230%
Chance of 1 Events/Landmarks: 25.661%
Chance of 2 Events/Landmarks: 25.070%
Chance of 3 Events/Landmarks: 17.541%
Chance of 4 Events/Landmarks: 9.816%
Chance of 5 Events/Landmarks: 4.657%
...
Chance of 2+ Events/Landmarks: 60.109%
Chance of 4+ Events/Landmarks: 17.499%
To answer the Colony/Dominate question: The probability of Colony is 25/260 (~9.6%); the probability of Dominate is about ~2.7% (limiting to 2 total Events/Landmarks), ~3.8% (without limiting).
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For example, there are 3 Gathering cards, so the chance that none are included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Wait, that's not right...? The chance for none of a 3 card set to be in any given game has to be way higher than that. How often do you have games that don't include any of Kings Court, Goons, and Grand Market?
He mixed up the explanation of the results. The ~11.1% is the chance that at least one of the cards in the set of 3 is in the Kingdom. My gut tells me that probability is close to reality in my total sample space of games.
Yep, my mistake.
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I want to know what the probability is of seeing any Event/Landmark.
Limiting to 2 total events/landmarks, the chance of any particular event/landmark is 2.7%.
Without limiting to 2, I get:
Chance of 0 Events/Landmarks: 14.230%
Chance of 1 Events/Landmarks: 25.661%
Chance of 2 Events/Landmarks: 25.070%
Chance of 3 Events/Landmarks: 17.541%
Chance of 4 Events/Landmarks: 9.816%
Chance of 5 Events/Landmarks: 4.657%
...
Chance of 2+ Events/Landmarks: 60.109%
Chance of 4+ Events/Landmarks: 17.499%
To answer the Colony/Dominate question: The probability of Colony is 25/260 (~9.6%); the probability of Dominate is about ~2.7% (limiting to 2 total Events/Landmarks), ~3.8% (without limiting).
The full table of the probabilities for number of Events vs. number of Landmarks, if not limited.
https://docs.google.com/spreadsheets/d/1dqtsMagkAqWIB1a-IzF-R7tY69QhPi4M3cOhHdbN5fU/
If you limit the total number, the expected number of Events+Landmarks is:
1.46 (max 2)
1.81 (max 3)
1.98 (max 4)
2.11 (no max)
If not limiting, the probability of seeing a particular Event or Landmark is exactly 10/261 (~3.8%).