Dominion Strategy Forum
Dominion => Dominion General Discussion => Topic started by: JW on March 02, 2014, 02:23:55 pm
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It seems like the chance of getting your Urchins to collide on turns 3-4 after a double Urchin opening should be a standard reference, but I don't see it anywhere (including the Wiki page (http://wiki.dominionstrategy.com/index.php/Urchin)).
I calculate this as 5/11 = 45.5% 5/12 .
Method: Without loss of generality, consider the first Urchin. There's an 11/12 chance that it is not the bottom card of the shuffle (in which case it can't collide). If it's cards 1-5 or 7-11, then there are 5 places that the second Urchin could be so they collide (e.g., card 11 collides with cards 6-10, card 1 collides with cards 2-6, etc.).
Edit: There's a special case that I initially missed that dnkywin points out where if the first Urchin is in card 6, then 10 of the remaining 11 positions collide with it (all but the bottom card) instead of 5 for the other locations.
So the chance is (10/12) * 5/11 + 1/12 * 10/11= 5/11.
If you don't draw any Urchins on turn 3, then by the same logic as above you have a 70.4% (5/7) chance to collide the Urchins on turn 4 (6/7 * 5/6= 5/7). There is a 1.52% (2/12 * 1/11) chance that your Urchins are at the bottom of your shuffle, in which case you know on turn 4 that you will collide them on turn 5. Otherwise, if your Urchins don't collide you'll presumably want to add a third Urchin before the reshuffle to increase the odds of collision on the next shuffle.
Thoughts? Anyone confirm or disagree with the calculations?
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Thoughts? Anyone confirm or disagree with the calculations?
pretty sure it's correct
e: no, dnkywin is right. for 12 > u ≠ 6 it's 5 possibilites, but for u = 6 it's 10.
ok, but i'm pretty sure that was the only mistake
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It's 5/11 = 0.45454545... -- looking at the positions of the Urchins in your deck, they will only collide if they're both in cards 1-6 (15 possibilities) or they're both in cards 6-11 (another 15 possibilities), giving 30/(12C2) = 30/66 = 5/11. (Where we disagree is that if the first urchin is Card 6, then the second urchin can be anywhere but the bottom for the two Urchins to collide. )
Your other probabilities look right.
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It's 5/11 = 0.45454545... -- looking at the positions of the Urchins in your deck, they will only collide if they're both in cards 1-6 (15 possibilities) or they're both in cards 6-11 (another 15 possibilities), giving 30/(12C2) = 30/66 = 5/11. (Where we disagree is that if the first urchin is Card 6, then the second urchin can be anywhere but the bottom for the two Urchins to collide. )
Your other probabilities look right.
Good catch, thanks.