Dominion Strategy Forum
Dominion => Dominion General Discussion => Topic started by: pst on May 03, 2013, 05:17:39 am
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[This topic is taken from here (http://forum.dominionstrategy.com/index.php?topic=7983.msg236111#msg236111). ]
So you want to get an early Familiar? Probably your first two buys are Potion and something else. Maybe that other card is Silver, maybe something even better. This table shows you your chances of getting that Familiar in turn 3 or 4 for some choices.
| Card | | t3 | | t4 | | t3+t4 |
| Squire | | 15.15% | | 15.15% | | 30.30% |
| Squire + draw | | 18.94% | | 18.94% | | 37.88% |
| Squire + Hermit | | 15.15% | | 41.67% | | 56.82% |
| Silver | | 32.70% | | 32.70% | | 65.40% |
| Poor House | | 32.83% | | 32.83% | | 65.66% |
| Lighthouse | | 30.05% | | 35.73% | | 65.78% |
| Sage | | 39.77% | | 27.15% | | 66.92% |
| double Hermit | | 0.00% | | 68.18% | | 68.18% |
| Moat | | 38.51% | | 38.51% | | 77.02% |
| Steward | | 39.65% | | 39.65% | | 79.29% |
| double Watchtower | | | | | | 98.48% |
Poor House is an example of a card that guarantees you enough money if you draw it with the Potion. The same goes for Beggar and Secret Chamber. So these are obviously slightly better than Silver.
Fishing Village is like Lighthouse ($1+$1) and turns out to be even better.
I've included the alternative way to get Familiar from Squire without a Potion. The first line for Squire is for when you get it with a trasher like Chapel that can trash Squire if you draw them together. The better line "Squire + draw" is for when your trasher draws a card first (Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
The idea can be improved with Squire + Hermit for easier finding of that Squire. In this case I assume you buy an additional turn 3 Squire if possible so your turn 4 Hermit can trash that one even if your first Squire is at the bottom of your deck. But with Hermit available (and a 4+3 opening) a much better opening is double Hermit and buy the Squire at turn 3! This only fails if you can't afford the Squire at turn 3 or if you don't get a Hermit in turn 4. If there is Watchtower and Squire then double Watchtower is a crazy good opening. With Watchtower in your hand buy a Squire that you immediately trash, and then topdeck the Familiar. If you get Watchtower without enough money in your first hand, then play it. The given probability is the probability that you get at least one Familiar by turn 4. It only fails if both of your Watchtowers miss the shuffle.
Moat is like Shanty Town, drawing two cards. Always play it.
And finally the best one-card option I've found is Steward. It is really good because you can use it for money if drawn with Potion, and for cards to get that Potion otherwise.
You always play Steward for cards if playing it for money isn't enough to buy a Familiar.
What else is there to say? I've assumed other players don't attack you or mess with you otherwise in a way that disturbs your chances.
I've done this by looking at all 12! permutations of your cards after your first two buys. I haven't looked at drawers like Warehouse when it's possible to draw beyond those 12 cards in turn 4. It's especially complicated since you can draw a card gained in turn 3 then.
There are also several cards that can't draw that much but that I still haven't considered anyway. Maybe I'll extend this later. Maybe someone else will.
Here is the same table again, but with number of permutations (out of those 12!) instead of percentages.
| Card | | t3 | | t4 | | t3+t4 |
| Squire | | 72,576,000 | | 72,576,000 | | 145,152,000 |
| Squire + draw | | 90,720,000 | | 90,720,000 | | 181,440,000 |
| Squire + Hermit | | 72,576,000 | | 199,584,000 | | 272,160,000 |
| Silver | | 156,643,200 | | 156,643,200 | | 313,286,400 |
| Poor House | | 157,248,000 | | 157,248,000 | | 314,496,000 |
| Lighthouse | | 143,942,400 | | 171,158,400 | | 315,100,800 |
| Sage | | 190,512,000 | | 130,032,000 | | 320,544,000 |
| double Hermit | | 0 | | 326,592,000 | | 326,592,000 |
| Moat | | 184,464,000 | | 184,464,000 | | 368,928,000 |
| Steward | | 189,907,200 | | 189,907,200 | | 379,814,400 |
| double Watchtower | | | | | | 471,744,000 |
Edit: Corrected Steward strategy. Added Moat/Shanty Town and double Watchtower.
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So did I do this correctly? I've double checked my first results, including Silver vs. Lighthouse, as well as my best result Steward, so these should be correct. As for the Squire options I just programmed them and ran them without doublechecking them, so possibly I've made an error there.
How did you come up with those numbers? I tried to run it in a break (okay, no sophisticated way of making sure there wasn't a mistake), but what I got was that:
The only way for FV(or equivalently lighthouse) to help you is if it is in your t3 hand *and* your t4 hand is C-C-E-E-P. The chances of this happening are 5 in 88.
The only way for FV to hurt you is if you have FV-C-E-E-P on turn 3 OR 4 (instead of having the silver). The chances of this happening are 7 in 66.
Ergo, you lose more than you gain, and your overall chances of getting familiar on t3-t4 decrease by 13 in 264, or a hair under 5%.
Anyone see a mistake with my math?
I don't know how you came up with those probs so I can't point out where the error is. It's easy to make blunders though, so that's why I sticked to looking at all permutations one by one. Somewhat dumb but simple! Here's my reasoning when doublechecking this result though:
X is the extra card (Silver or Fishing Village/Lighthouse).
Fishing Village is better when your t3 hand contains X and your t4 hand is CCPEE.
You can pick the two Coppers in that hand in C(7,2) ways and the two Estates in C(3,2) ways. And then X must be among the 5 cards in your t3 hand (and not in the 2 cards left), so you get C(7,2) * C(3,2) * 5/7 = 21*3*5/7 = 45 cases.
Fishing Village is worse when your t3 hand contains XPCEE. There 7 Coppers to choose from and C(3,2) Estates.
7*3 = 21. It's also worse when your t4 hand contains this, so all in all 42 cases.
So the net gain is 45-42 = 3 cases. For each "case" I can actually permute the five needed cards in any way, and the seven other cards in any way. 3*5!*7! = 1814400 which is the difference I have.
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How about Shanty Town/Moat?
I've always thought Shanty Town was somewhat comparable to Silver in being able to afford Familiar as a hand of CPSEE can still net 2 Coppers unless your last Estate is badly placed and a hand of CCPSE will only fail if it draws exactly 2 Estates.
Why I liked Shanty Town better is because of the added cycling once you get that Familiar.
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With Steward but no Potion in your turn 3 hand you should draw cards if you have 4 Coppers, otherwise not.
Why?
If I have already 4 coppers in hand and draw 2 cards of my remaining 7, there is a bigger chance for drawing a fifth copper of my remaining 3 (leaving 2$P for t4) than for drawing that single potion. Maybe this is too intuitive a question to hold against your statistics, but I'd like to know, nonetheless.
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I'd love to know the statistics for Storeroom and Courtyard.
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I'd love to know the statistics for Storeroom and Courtyard.
This. These should be the strongest ones.
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With Steward but no Potion in your turn 3 hand you should draw cards if you have 4 Coppers, otherwise not.
Why?
If I have already 4 coppers in hand and draw 2 cards of my remaining 7, there is a bigger chance for drawing a fifth copper of my remaining 3 (leaving 2$P for t4) than for drawing that single potion. Maybe this is too intuitive a question to hold against your statistics, but I'd like to know, nonetheless.
Because you're very unlikely to get a Familiar on T4 in that case as that would require 2 of your 3 Estates (which are all still in the deck) to be on the bottom, which is less likely than your Potion being one of the two cards you'd draw with Steward.
This is different when your hand is Steward + 3 Coppers + Estate, as now only 2 Estates remain in your deck, so that one Estate + 1 Copper on the bottom would also get you a T4 familiar. Now drawing 2 cards has a higher probability of ruining a T4 familiar than getting a T3 one.
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pst, how did you calculate Steward?
Let's say you have the order of cards is SEECCPECCCCC.
Either way you don't get a Familiar in turn 3, but if you can buy a Familiar in turn 4 depends on what you choose, but this is knowledge you don't have.
With SEECCPECCCCC you shouldn't draw, but it could also be SEECCECCCCCP where you should definitely choose to draw.
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I suppose you could just enumerate all possibilities (with draw and $2) and work out backwards what you should have chosen.
So you start from SEECC and enumerate all possibilities for drawing 2 and getting $2 (or even trashing) and decide whether there are more good results in drawing or not drawing and pick that as the default move.
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A note about Courtyard is that it's not as good as comments here would indicate, as often when you get Familiar on turn 4, you will have triggered a reshuffle already, which is kinda ugly (all of Potion, Familiar and Courtyard won't be in your deck).
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Opening Watchtower/Watchtower on a Squire board and you're guaranteed a Familiar, unless attacked or both Watchtowers miss the reshuffle.
Edit: Oh, one other failure is if you draw WEEEC on T3 and then CCCCC on T4. But CCCCC on T3 and WEEEC on T4 still gets a Familiar, since you can play your Watchtower T4 and know you will draw another Watchtower and a Copper.
On third thought, if you draw WEEEC on T3, then play Watchtower. Either you draw WC now and get your familiar, or you draw CC and guarantee that your T4 hand is WCCCC.
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By the way, the double Watchtower opening has a 98.48% chance of getting your a familiar before the next shuffle (excluding player interaction)
The better question, in fact, is what the likelihood of getting two Familiars by the second reshuffle is.
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw. Masquerade draws 2 and trashes, and is probably the best bet here, though Masq + Potion might have an even higher probability.
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pst, how did you calculate Steward?
Without any secret knowledge, but just playing as I wrote:
With Steward but no Potion in your turn 3 hand you should draw cards if you have 4 Coppers, otherwise not.
(All other choices are obvious.)
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pst, how did you calculate Steward?
Without any secret knowledge, but just playing as I wrote:
With Steward but no Potion in your turn 3 hand you should draw cards if you have 4 Coppers, otherwise not.
(All other choices are obvious.)
Ah, I thought this was the result of the computation. It was rather the assumption for your computation. Now I understood it.
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Forager doesn't draw.
Oops, thanks. I'm fixing that in the OP. I'm still so unused to DA that when I happened to get that wrong first I wouldn't notice it later.
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
Nope. Overgrown Estate draws, Forager does not.
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
Nope. Overgrown Estate draws, Forager does not.
Actually, you draw, the cards do not.
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Actually, you draw, the cards do not.
But, in Soviet Russia...
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
Nope. Overgrown Estate draws, Forager does not.
Actually, you draw, the cards do not.
either way, it still increases the chance of getting a familiar. Just never count on it...
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
Nope. Overgrown Estate draws, Forager does not.
Actually, you draw, the cards do not.
either way, it still increases the chance of getting a familiar. Just never count on it...
No, it doesn't, because the card you're trashing should be a Squire, not an Overgrown Estate. Unless you went for Potion/Forager, which isn't a great opening on a Familiar board.
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I find it amusing that the combos that do better than Silver/Potion require either:
One of two three-card combos (1 in what, 1000 games?);
Opening Potion/X anyway.
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I think a better question is what is the likelihood of getting any $5 or $6 attack card using Squire and a trasher, as opposed to opening Silver/X. Having Squire/[Hermit|Watchtower]/[any $5/$6 attack] on the board is many times more likely than Squire/trasher/Familiar.
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How about Shanty Town/Moat?
Added!
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By the way, the double Watchtower opening has a 98.48% chance of getting your a familiar before the next shuffle (excluding player interaction)
Thanks, right! And you can topdeck them to boot. (Added to the table.)
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With Steward but no Potion in your turn 3 hand you should draw cards if you have 4 Coppers, otherwise not.
Why?
If I have already 4 coppers in hand and draw 2 cards of my remaining 7, there is a bigger chance for drawing a fifth copper of my remaining 3 (leaving 2$P for t4) than for drawing that single potion. Maybe this is too intuitive a question to hold against your statistics, but I'd like to know, nonetheless.
Because you're very unlikely to get a Familiar on T4 in that case as that would require 2 of your 3 Estates (which are all still in the deck) to be on the bottom, which is less likely than your Potion being one of the two cards you'd draw with Steward.
This is different when your hand is Steward + 3 Coppers + Estate, as now only 2 Estates remain in your deck, so that one Estate + 1 Copper on the bottom would also get you a T4 familiar. Now drawing 2 cards has a higher probability of ruining a T4 familiar than getting a T3 one.
I did this wrong and have corrected the original post now. If the $2 isn't what you need it's never wrong to draw cards instead! (Only thinking about the getting-Familiar-probability, of course) If you have Steward, Copper, 3 Estates in your t3 hand it doesn't matter, but otherwise it's always to better to draw in t3 if you have Steward without Potion.
It's easy to see that it doesn't matter in that case. It's not possible to buy a Familiar in t3 because even if you draw the Potion you won't afford it. So all your hope is to t4, and you have the same chance to achieve that with the top five cards as with the bottom five cards of your deck.
If you have at least 2 Coppers in your hand the same reasoning applies, except now you have a chance to get the Familiar in turn 3 if you draw, so it's a win.
I'm sorry for doing this wrong at first!
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The better line "Squire + draw" is for when your trasher draws a card first (Forager, Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
Forager doesn't draw.
if you trash your overgrown estate it does
Nope. Overgrown Estate draws, Forager does not.
Actually, you draw, the cards do not.
either way, it still increases the chance of getting a familiar. Just never count on it...
No, it doesn't, because the card you're trashing should be a Squire, not an Overgrown Estate. Unless you went for Potion/Forager, which isn't a great opening on a Familiar board.
true. ah well, I tried
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On the other hand, in a game where people are opening Watchtower/Watchtower, Familiar is usually going to be "+1 card, +1 action, trash a Curse from the supply"—how badly do you really want that Familiar?
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On the other hand, in a game where people are opening Watchtower/Watchtower, Familiar is usually going to be "+1 card, +1 action, trash a Curse from the supply"—how badly do you really want that Familiar?
But if you don't want that Familiar, you don't want double Watchtower. And if you don't have double Watchtower, your opponents will want that Familiar.
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I find all of the squire-based lines confusing - particularly anything that doesn't specifically have squire with a trasher - but also reasonably useless compared to other things.
I recalculated things from yesterday and am now getting figures agreeing with yours - not sure what I did wrong.
Anyway, here's some other possibilities of interest:
Potion/nothing: 60.606% chance
Potion/copper: 60.101% chance (Obviously this extends to things like squire as well that give you $1)
Potion/(Warehouse/courtyard): 93.434% chance
Then the interesting one - herbalist. Playing where you only return a potion from t3 to t4 if possible bumps your chances up to 62.88%. However, you get very slightly better chances by returning copper if you have a t3 herbalist without potion and with 3 or more coppers. This lifts you up to a 63.51% chance. Furthermore, in either case, there is a 7.72% chance of getting TWO familiars on turns 3-4. So, if you're playing to maximize the number of familiars you have at the end of t4, with herbalist you expect .7123.
Also, if you think about it, oracle is going to be VERY high, as it can do everything moat can plus a little more. Haven't sat down to calculate this, though, as it would be really complicated in some cases (triggering a reshuffle before the end of t4 leads to needing more information, at the least).
Finally, courtyard/potion going for *golem* does not seem at all an unreasonable play.
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I find all of the squire-based lines confusing - particularly anything that doesn't specifically have squire with a trasher - but also reasonably useless compared to other things.
I've included the alternative way to get Familiar from Squire without a Potion. The first line for Squire is for when you get it with a trasher like Chapel that can trash Squire if you draw them together. The better line "Squire + draw" is for when your trasher draws a card first (Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
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I find all of the squire-based lines confusing - particularly anything that doesn't specifically have squire with a trasher - but also reasonably useless compared to other things.
I've included the alternative way to get Familiar from Squire without a Potion. The first line for Squire is for when you get it with a trasher like Chapel that can trash Squire if you draw them together. The better line "Squire + draw" is for when your trasher draws a card first (Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
I understand that - I did read what he put there - but it's not intuitive to just look at watchtower/watchtower, for instance, and see aha, I am getting a familiar. If it said "Double watchtower (squire)", it would be much better IMO.
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Also, if you think about it, oracle is going to be VERY high, as it can do everything moat can plus a little more. Haven't sat down to calculate this, though, as it would be really complicated in some cases (triggering a reshuffle before the end of t4 leads to needing more information, at the least).
And it sometimes screws up your opponent's successful Familiar purchase by discarding the Potion or two Coppers or their Oracle (or whatever they bought with their $3).
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I find all of the squire-based lines confusing - particularly anything that doesn't specifically have squire with a trasher - but also reasonably useless compared to other things.
I've included the alternative way to get Familiar from Squire without a Potion. The first line for Squire is for when you get it with a trasher like Chapel that can trash Squire if you draw them together. The better line "Squire + draw" is for when your trasher draws a card first (Rats, Upgrade, Junk Dealer). Both of these are much worse than Potion + Silver.
I understand that - I did read what he put there - but it's not intuitive to just look at watchtower/watchtower, for instance, and see aha, I am getting a familiar. If it said "Double watchtower (squire)", it would be much better IMO.
The best way to cue a dominion player to non-standard gaining is with *.
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The asterisk is a good idea.
Also, the "Draw" terminology is really confusing. I was sitting there wondering how squire+smithy gets you a familiar.
I would suggest:
1) Squire+Trash from hand
2) Squire+Cantrip Trasher
3) 2x Watchtower (with squire)
for those cases. Better to be explicit with it.
Actually, since you have space, you might as well make the other columns "P+Silver, P+Moat", etc. Those lines are all short anyway.
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How about Shanty Town/Moat?
Added!
Great, now we can finally conclude that Moat is better than Silver.
Moat, passing the Silver test since Alchemy.
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On the other hand, in a game where people are opening Watchtower/Watchtower, Familiar is usually going to be "+1 card, +1 action, trash a Curse from the supply"—how badly do you really want that Familiar?
But if you don't want that Familiar, you don't want double Watchtower. And if you don't have double Watchtower, your opponents will want that Familiar.
So it sounds like you should open double-Watchtower anyway, for flexibility? If your opponent doesn't go Watchtower, you go for Familiar; if they do, well, at least you're defended if they get a Familiar?
If your opponent goes double-Watchtower, the thing to do is probably wait to get a Familiar until their deck is a bit more padded, I guess.
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Squire/Watchtower also works as an engine, so those watchtowers aren't wasted either way.
Still takes some trashing, those coppers will get in the way of squire/watchtower... but if there isn't any trashing, you'll want to get familiars anyway, because then even a small number of curses that get through the defenses are a big deal. Maybe not immediately, but at some point.
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On the other hand, in a game where people are opening Watchtower/Watchtower, Familiar is usually going to be "+1 card, +1 action, trash a Curse from the supply"—how badly do you really want that Familiar?
But if you don't want that Familiar, you don't want double Watchtower. And if you don't have double Watchtower, your opponents will want that Familiar.
So it sounds like you should open double-Watchtower anyway, for flexibility? If your opponent doesn't go Watchtower, you go for Familiar; if they do, well, at least you're defended if they get a Familiar?
If your opponent goes double-Watchtower, the thing to do is probably wait to get a Familiar until their deck is a bit more padded, I guess.
I don't know. At that point, you might as well just keep buying Watchtowers, right?
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If your opponent goes double-Watchtower, the thing to do is probably wait to get a Familiar until their deck is a bit more padded, I guess.
I don't know. At that point, you might as well just keep buying Watchtowers, right?
Depends on whether there's a good Watchtower-based engine to be built, I guess. Otherwise you're loading up on terminals and not actually building your deck.
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If your opponent goes double-Watchtower, the thing to do is probably wait to get a Familiar until their deck is a bit more padded, I guess.
I don't know. At that point, you might as well just keep buying Watchtowers, right?
Depends on whether there's a good Watchtower-based engine to be built, I guess. Otherwise you're loading up on terminals and not actually building your deck.
Edge case: Sarcasm.
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What about Squire+Masquerade?
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What about Squire+Masquerade?
I don't want a Familiar with Masquerade on the board.
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What about Squire+Masquerade?
I don't want a Familiar with Masquerade on the board.
Sure you do. At least you'd want one for free, right? Or six of them for free?
Or do you mean that you wouldn't (go to hell, edge-casers) bother with the nuisance of a Potion and the risk of catching an early 2+P?
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Well, it's still not free, you spent a buy on a squire which you're going to trash AND you spent one masquerade turn on trashing that squire instead of playing it and trashing an estate. So it puts you pretty substantially behind at first, it's still an investment. And you can't even get more than one familiar this way, not without spending even more time buying squires and getting them to collide with masquerades.
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What about Squire+Masquerade?
I don't want a Familiar with Masquerade on the board.
Sure you do. At least you'd want one for free, right? Or six of them for free?
Or do you mean that you wouldn't (go to hell, edge-casers) bother with the nuisance of a Potion and the risk of catching an early 2+P?
It's a card that Masquerade can draw dead, and at best it gives out curses, which will be passed back to you, slowing down your own trashing prowess. It's a pretty big conflict.
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Well, it's still not free, you spent a buy on a squire which you're going to trash AND you spent one masquerade turn on trashing that squire instead of playing it and trashing an estate. So it puts you pretty substantially behind at first, it's still an investment. And you can't even get more than one familiar this way, not without spending even more time buying squires and getting them to collide with masquerades.
$2 + buy + untrashed estate/copper is nothing to sneeze at, but keep that cost in perspective.
In a kingdom without Masquerade, Familiar is so strong that you're typically willing to open Potion + Silver to get one. You spend your $4 open on a card that will be useless for most of the game and you still face a 35% chance of missing it on the first shuffle or even drawing $2+P. By contrast, $2+Squire is perfectly fine... and whether you get your Familiar or not, you still end the first shuffle 1-2 cards slimmer than if you'd opened Potion.
The real question is how effectively Masquerade counters Familiar. If you draw a Curse with your Masquerade, you'll probably swap it for a Copper or an Estate. So when things go well, your opponent's Familiar has effectively handed you a Copper.
I'll concede that this counter is decent enough in a Masquerade-BM deck. A Copper instead of a Curse can mean buying a Gold instead of a Silver, and you don't really want to slim down aggressively anyway. But in an engine deck, Masquerade hardly counters Familiar at all. Being handed a Copper is very nearly as bad as being handed a Curse, and being handed 5-10 over the course of a game is devastating. When your Masquerade is up against an enemy Familiar, it makes no net progress slimming your deck at all.
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What about Squire+Masquerade?
I don't want a Familiar with Masquerade on the board.
Well, you don't want to do a lot of things whose probabilities were computed for this thread, so that's not really the point. Squire+Masq is pretty easy to compute. You need the Masq to show up on turn 3 or 4, and the Squire to be one of the other 6 cards that show up with it. So the probability is 10/12*6/11 = 5/11 = 0.4545...
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I think that I've beaten a somewhat mediocre Familiar deck with Masq-BM before. And if I can build any kind of decent engine, I might just skip Familiar.
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I get very annoyed when my opponent opens Potion + Scheme and still gets a Familiar on the next shuffle!
So what are the odds for this? If I open Silver it's a 65.40% chance for me. And a 60.606% chance for the player who opened Scheme?
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I get very annoyed when my opponent opens Potion + Scheme and still gets a Familiar on the next shuffle!
So what are the odds for this? If I open Silver it's a 65.40% chance for me. And a 60.606% chance for the player who opened Scheme?
Yes. The thing is, while of course you're far less likely to draw 2p or worse, you're also less likely to draw the potion at all (he has a 1 in 11 chance, you have a 1 in 6), which offsets most of that advantage.
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I computed the probability for Potion + Oasis, because I thought it would be quite high. I'm not sure if I computed it the correct way, but I got 72.1%. This is a bit higher than silver, as expected, since the probability of your Potion missing the reshuffle is quite low. I assumed that you always play Oasis, except when you draw Oasis, 3 Estates, Copper on turn 3.
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I get very annoyed when my opponent opens Potion + Scheme and still gets a Familiar on the next shuffle!
So what are the odds for this? If I open Silver it's a 65.40% chance for me. And a 60.606% chance for the player who opened Scheme?
Yes. The thing is, while of course you're far less likely to draw 2p or worse, you're also less likely to draw the potion at all (he has a 1 in 11 chance, you have a 1 in 6), which offsets most of that advantage.
Right, I see now. Now I'm unsure what the best of those openings are. I'm leaning towards the Silver anyway, for a better long-term deck, and then get Schemes for $3 after that.
(Btw, the numbers you gave were of course the odds of NOT drawing the Potion...)
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So I posted this in the Doctor preview but didn't get any response. I think Doctor/Squire with Familiar on the board is a little interesting.
If you have a 2/5 opening, buy Squire with 2C, discard 3E,2C,S, draw 5C. Buy Doctor with 5, shuffle 3E,2C,S into deck, turn over and trash two. The chance of one of those being a Squire is 1-(2/3) = 1/3. Then you get a Familiar in your discard pile, so your draw deck is 4 cards (copper/estate) and your discard pile is 5C, D, F. You then have a 1/7 chance of drawing Familiar on the third turn.
Well, considering you need 2/5 and not 5/2 for this, and it's only a 1/21 chance given the 2/5 opening, it's not all that likely. But it does let you play the Familiar on turn 3, which is neat. Plus, you've already trashed away one of your Estates and Coppers and can start naming Estate or Copper with Doctor. Seems like you'd be in a really sweet spot if it hit.
You can also do it with 4/3 opening, but then you cut your chance of trashing the squire in half. But that still lets you get a Familiar in your deck on Turn 3 1/6 of the time. Having a coin token from Baker of course makes this better.
If you miss trashing from Doctor's on-buy ability, you can still play Doctor to trash. To trash the Squire on turn 3, you'd have to draw the Doctor and have the Squire be cards 6, 7 or 8 in your reshuffle (draw 1st 5, reveal next 3). The number of ways to do this would be, (if you trashed two cards with Doctor previously), I think, 3/20. (Doctor must be in first 5 cards (1/2), Squire must be in last 5 cards (1/2), squire must be in first 3 of last 5 cards (3/5). So, 15%.
Now if you draw Doctor turn 3 but miss Squire, then you're screwed, because you know you'll be drawing Squire next turn. Or if you have $5 you can overpay $2 for Doctor and guarantee to trash the Squire, but then you'll have two Doctors.
Anyway, that's not all the possible cases, but I wonder how viable opening Doctor/Squire with Familiar is. You could do this for other attacks if you're unable to buy them in the opening, I guess, but it's probably better to just buy money and directly buy the attacks.
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With Baker in the setup and opening Potion/whatever you'll have a better chance to get a Familiar.
So long as the whatever gives at least 1 coin.
The only way to miss is Copper-Potion-3E
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You can still draw Copper/Estate/Estate/Estate/Potion on turn 5.
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So I looked into Squire/2xWatchtower again.
Probability of 0 Familiars: 1/66 = 1.5%
Probability of 1 Familiar: 125/198 = 63.1%
Probability of 2 Familiars: 35/99 = 35.4%
So the expected value is 265/198 = 1.34 Familiars.
Perhaps also of interest:
Probability of topdecking a t3 Familiar: 5/33 = 15.2%
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I just opened Potion/Storeroom and played my Familiar on turn 4. Storeroom seems good because it guarantees $3+P if you draw it with (or it draws) your Potion; it's also nice to have around after you bought the Familiar, because it cycles quickly.
If you draw Storeroom with 2 or 3 Estates on T3, play it, cycling exactly two cards - this guarantees a T3 or T4 Familiar, either you hit your Potion, or your next hand will be at least $3+P. On T4, don't trigger a reshuffle with your Storeroom, unless you drew your Potion on T3 without $3, in which case cycle all the way, you need to hit that Potion again. Otherwise, cycle as much as possible (except your Potion, obviously). If you cycle all the way on T3 but don't hit your Potion, buy another Storeroom*.
*If you really want to maximise your chances of getting that Familiar, that is. It would be better play to buy Silver (or equivalent), but in that case where your hand is Potion/Estate/Estate and then two cards from the rest of your deck, Storeroom is a guaranteed Familiar while Silver can be drawn with your one other Estate. The probabilities are practically identical, but this thread seems to be about maximising that probability, not finding the best whole-game strategy.
(St/C/E/P) = Storeroom/Copper/Estate/Potion
nCr = n Choose r
F = guaranteed T3/T4 Familiar, NF = guaranteed no T3/T4 Familiar
Here's my probability tree:
- Storeroom on T3: (5/12)
-- St/P/x/x/x (4/11) F
-- St/C/E/E/E or St/C/C/E/E ((7C1 + 7C2*3C2)/11C4 = 7/33) F
-- St/C/C/C/E or St/C/C/C/C: ((7C3*3C1 + 7C4)/11C4 = 14/33)
--- Storeroom hits Potion (4/7) F
--- Storeroom doesn't hit Potion: (3/7)
---- (34/45 F, 11/45 NF - see below...)
- C/C/C/C/P or C/C/C/E/P on T3 ((7C4 + 7C3*3C1)/12C5 = 35/198) F
- C/C/E/E/P or C/E/E/E/P on T3: ((7C2*3C2 + 7C1)/12C5 = 35/396)
-- Storeroom on T4 (5/7):
--- Hits Potion (2/9) F
--- Misses Potion (7/9) NF
-- No Storeroom on T4 (2/7) NF
- No Storeroom or Potion on T3: (7/12 * 6/11 = 7/22)
-- Storeroom on T4 (5/7) F
-- No Storeroom on T4: (2/7)
--- Potion on T4: (5/6)
---- C/C/C/C/P or C/C/C/E/P ((7C4 + 7C3*3C1)/10C4 = 2/3) F
---- C/C/E/E/P or C/E/E/E/P ((7C2*3C2 + 7C1)/10C4 = 1/3) NF
--- No Potion on T4 (1/6) NF
T3 was three times as likely to have been St/C/C/C/E than St/C/C/C/C. This affects the chances of your last three cards being P/C/C, P/C/E or P/E/E.
...
- Draw a Storeroom on T4 (1 - 8C2/10C2 = 17/45) F
- No recycled Storeroom on T4: (8C2/10C2 = 28/45)
-- T3 was St/C/C/C/C: (1/4)
--- Remainder of deck is P/E/E (3C2/6C2 = 1/5) NF
--- Remainder of deck is P/C/E: ((3C1*3C1)/6C2 = 3/5)
---- Draw C/C (6C2/8C2 = 15/28) F
---- Don't (13/28) NF
--- Remainder of deck is P/C/C: (3C2/6C2 = 1/5)
---- Draw C (1 - 3C2/8C2 = 25/28) F
---- Draw E/E (3C2/8C2 = 3/28) NF
-- T3 was St/C/C/C/E: 3/4
--- Remainder of deck is P/E/E (1/6C2 = 1/15) NF
--- Remainder of deck is P/C/E: ((2C1*4C1)/6C2 = 8/15)
---- (further probabilities as before)
--- Remainder of deck is P/C/C: (4C2/6C2 = 2/5)
---- (further probabilities as before)
Putting all of that together, I get a 3089/3564 = 86.67% chance of a T3/T4 Familiar. Feel free to check my working, I'm not inerrant. Sometimes you get to buy it on T3 and play it on T4; sometimes you can even buy a second Familiar on T4 in this situation. Sometimes you buy on T4, but you forced a reshuffle so there are still 7 cards left in your deck - your Storeroom is in there to assist cycling, though.
As expected, Storeroom is one of the better openings for getting a Familiar; you will buy your Familiar on T3/T4 in about 8 games out of 9.
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3.) Potion cards
The best use for Storeroom are the combinations with Potion cards. The problem with Potion is that you want to see it often early in the game but don't want to clog your deck with 2 Potions. Also your Potion(s) are dead cards later in the game. Stategies with Potion cost cards also need +Buy for cases when you hit $5P or better. Your supporting card also should cost less than $4 that you can open Potion+X. All this solves Storeroom. You always want to open Storeroom + Potion and discard all 4 cards when you don't have a Potion in hand. The probability is high to draw your Potion which guarantees you $3P which is enough for 8 out of 10 Potion cards. If you miss your Potion you should have at least $3 to buy a second Storeroom.
3c) Familiar
Like written above a Storeroom and a Potion guarantees you a $3P hand, exactly the amount you need for Familiar. The probability is therefore high you can buy a Familiar in turn 3 or 4 with this opening. Storeroom also triggers the reshuffle more often which you need to play your Familiar as often as you can early in the game. The problem is that you can't play a drawn Familiar because Storeroom is terminal, so take that into account.
Nice to have some statistic data to confirm this. Thanks.
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Because you can occasionally pick up another Familiar on T4 after already buying one on T3, let's calculate the probability for that, too.
FF = double T3/T4 Familiar.
- Storeroom on T3: (5/12)
-- St/P/x/x/x, or St/C/C/C/E or St/C/C/C/C hitting Potion: (4/11 + (14/33)(4/7) = 20/33) F
--- T4 draw Familiar: (2/10 = 1/5) [Curse on T4 with (5/12)(20/33)(1/5) = 5/99 probability]
---- With Storeroom: (2/9)
----- Draw with or hit Potion (5/8) FF
--- Don't draw T4 Familiar: (8/10 = 4/5)
---- Draw Storeroom: (2/9)
----- Draw with Potion (1/8) FF
----- Hit Potion (4/9) FF
- C/C/C/C/P or C/C/C/E/P on T3 ((7C4 + 7C3*3C1)/12C5 = 35/198) F
-- Storeroom on T4: (5/7)
--- Storeroom hits Potion: (2/11) FF
Giving an overall 19595/352836 = 5.55% chance of buying Familiar on both T3 and T4.
Summary: You get your T3/T4 Familiar 86.67% of the time (42.93% for just T3), you give a Curse on T4 5.05% of the time, and you get two Familiars 5.55% of the time. These aren't conditional probabilities, that's a 5.55% absolute probability that opening Storeroom/Potion gets you two Familiars by turn 4.
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You always want to open Storeroom + Potion and discard all 4 cards when you don't have a Potion in hand.
This isn't true; if you draw 2 or 3 Estates with your T3 Storeroom, cycle for exactly two cards - if you don't draw your Potion, your next hand will be Potion and at least three Coppers. Also if you draw Storeroom on T4 but didn't see your Potion yet, cycle exactly two cards so that when you buy your Familiar, it won't miss the reshuffle.
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Sage/Potion is also quite strong to ensure an early Familiar. With your Sage, you've basically 2 Potions in your deck, because as long as you don't buy something different than Sages or Potions, it will always stop at a potion. If you hold $2+Sage, you naturally don't play Sage. If you miss it anyway, you're able to search your potion very quickly. When finally got familiar, Sage is super-helpful to play it often. The only problem is that you've no money density and basically begin to build up your deck after the curse pile is empty, so the whole thing is basically a bad idea if you cannot weaken your opponent enough (e.g. Curse Counters)
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Sage seems OK but not great, it does let you play your Familiar more often too once you have it ($3+P >= $3), but you're setting yourself up for hands where you draw Sage with your Potion. The problem with getting a Familiar isn't that you don't have enough Potions, but rather that you don't draw enough money with your Potion, and Sage doesn't solve that problem except by giving you another "virtual" Potion which can be drawn with $3 if your first one wasn't. Let's calculate: (These probabilities will, of course, be the same for a Potion/Potion opening with Baker on the board, not that you would ever do that...)
Notation: C = Copper, E = Estate, P = Potion (or Sage)
- Potion (or Sage) on T3: (1 - 10C5/12C5 = 15/22)
-- T3 C/C/C/P/P (7C3/11C4 = 7/66) F
-- T3 C/C/C/C/P (7C4/11C4 = 7/66) F
-- T3 C/C/C/E/P (7C3*3C1/11C4 = 7/22) F
-- T3 C/C/E/E/P: (7C2*3C2/11C4 = 21/110)
--- T4 C/C/C/C/P (5C4/7C5 = 5/21) F
--- T4 C/C/C/E/P (5C3/7C5 = 10/21) F
-- T3 C/E/E/E/P: (7C1*3C3/11C4 = 7/330)
--- T4 C/C/C/C/P (5/7) F
- No Potion on T3: (10C5/12C5 = 7/22)
-- Potion on T4: (1 - 1/7C5 = 20/21)
--- T4 C/C/C/P/P (7C3/11C4 = 7/66) F
--- T4 C/C/C/C/P (7C4/11C4 = 7/66) F
--- T4 C/C/C/E/P (7C3*3C1/11C4 = 7/22) F
That gives me 2725/4356 = 62.56% chance of a T3/T4 Familiar, worse than Silver. Let me know if there's a mistake.
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Courtyard stats were requested, so I'll have a go at that too. At first it seems strong because you can put your Potion back, but then you realise it's stronger - you get a 7-card hand which, if it contains your Potion, must still contain at least three Coppers. If you don't hit your Potion, obviously put a Copper back, not an Estate...
There is the problem of triggering a reshuffle when you play Courtyard on T4. You can buy a Familiar, but it will take about an extra turn before you can play it - this is denoted Fx.
If you draw $2+P on T3, don't buy another Courtyard - you need Courtyard to grab the Potion from the reshuffle on T4, which is 1/5 if you don't put another card there, but 1/6 if you do.
- Draw Courtyard on T3: (5/12)
-- Courtyard hits Potion (7/11) F
-- Courtyard misses Potion: (4/11)
--- $3+P on T4 ((6C3 + 6C2*3C1)/9C3 = 65/84) F
- No T3 Courtyard: (7/12)
-- T3 Potion: (5/11)
--- $3+P on T3 ((7C4 + 7C3*3C1)/10C4 = 2/3) F
--- no $3+P: ((7C2*3C2 + 7C1)/10C4 = 1/3)
---- T4 Courtyard hits reshuffled potion ((5/7)(1/5) = 1/7) Fx
-- No T3 Potion: (6/11)
--- T4 Courtyard (5/7) Fx
--- No T4 Courtyard: (2/7)
---- Potion: (5/6)
----- $3: ((7C4 + 7C3*3C1)/10C4 = 2/3) F
Giving 785/924 = 84.96% chance of a T3/T4 Familiar buy. As expected, Courtyard is pretty reliable too.
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Potion/(Warehouse/courtyard): 93.434% chance
It seems one of us must be wrong - though I'm highly skeptical that Warehouse and Courtyard are identical. T3 Courtyard returns a Copper to your deck if it doesn't draw your Potion, leaving you with a five card hand for T4 guaranteed to contain your Potion and that Copper. On the other hand, T3 Warehouse missing the Potion leaves you with only four cards in your deck, so you will reshuffle and draw a card you already saw, which could even be Warehouse.
Any Potion/X opening (excluding on-buy/on-gain effects) has an 8.84% chance of drawing $1+P / $2+P on T3, which doesn't guarantee not seeing your Potion again on T4, but also surely isn't the only way it can fail.
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Potion/(Warehouse/courtyard): 93.434% chance
It seems one of us must be wrong - though I'm highly skeptical that Warehouse and Courtyard are identical. T3 Courtyard returns a Copper to your deck if it doesn't draw your Potion, leaving you with a five card hand for T4 guaranteed to contain your Potion and that Copper. On the other hand, T3 Warehouse missing the Potion leaves you with only four cards in your deck, so you will reshuffle and draw a card you already saw, which could even be Warehouse.
You are probably right. I don't have the paper I worked this up on anymore, so I can't remember what my thought process is, but having thought about it for 30 seconds, I don't find fault with your argument.
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You always want to open Storeroom + Potion and discard all 4 cards when you don't have a Potion in hand.
This isn't true; if you draw 2 or 3 Estates with your T3 Storeroom, cycle for exactly two cards - if you don't draw your Potion, your next hand will be Potion and at least three Coppers. Also if you draw Storeroom on T4 but didn't see your Potion yet, cycle exactly two cards so that when you buy your Familiar, it won't miss the reshuffle.
Thanks, I agree. What I meant to say is "cycle as much as possible" but haven't thought of edge cases, a big mistake whenever you talk about Dominion.
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That gives me 2725/4356 = 62.56% chance of a T3/T4 Familiar, worse than Silver. Let me know if there's a mistake.
Ok, the chance on t3/t4 are maybe not as good as silver, but with sage, it's less painful if you get it on t5, because you are shuffling more quickly. And you can play your familiar more and getting a 2nd faster, of course.
The main downside on Sage/Potion is, if you draw them together on t3, where you don't have any card which costs $3+ yet and so Sage is basically a dead card.
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The main downside on Sage/Potion is, if you draw them together on t3, where you don't have any card which costs $3+ yet and so Sage is basically a dead card.
Well, at least it puts your deck into your discard pile, so your Potion comes up again sooner.
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Oh, I did Potion/Warehouse but I forgot to post it here. It is much better than Silver, very nearly as good as Courtyard (~5/6) though the details are different; also, you would probably rather have the Warehouse around because if it draws your Familiar, you can play it that turn.
- T3 Warehouse: (5/12)
-- Hits Potion (7/11) F
-- Doesn't hit Potion: (4/11)
--- Draw reshuffled Warehouse on T4 (1/8) F
--- Don't: (7/8)
---- $3+P ((7C4 + 7C3*3C1)/10C4 = 2/3) F
- No T3 Warehouse: (7/12)
-- T3 Potion: (5/11)
--- $3+P ((7C4 + 7C3*3C1)/10C4 = 2/3) F
--- Not: (1/3)
---- T4 Warehouse (5/7) hits reshuffled Potion (1/5) Fx
-- No T3 Potion: (6/11)
--- T4 Warehouse (5/7) Fx
--- No T4 Warehouse: (2/7)
---- T4 Potion: (5/6)
----- $3+P ((7C4 + 7C3*3C1)/10C4 = 2/3) F
Giving a T3/T4 Familiar probability of 665/792 = 83.96%, pretty strong, one percentage point less than Courtyard.