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[WanderingWinder]

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.

[florrat]

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.

[Jeebus]

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...)

[Witherweaver]

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.

[Davio]

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

[Warfreak2]

You can still draw Copper/Estate/Estate/Estate/Potion on turn 5.

[SirPeebles]

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%

[Warfreak2]

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.

[Qvist]

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.

[Warfreak2]

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.

[Warfreak2]

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.

[achmed_sender]

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)

[Warfreak2]

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.

[Warfreak2]

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.

[Warfreak2]

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.

[WanderingWinder]

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.

[Qvist]

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.

[achmed_sender]

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.

[Drab Emordnilap]

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.

[Warfreak2]

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.