Opening Probabilities, Part II

Last time we examined some basic opening probabilities for Silver/Silver openings and Chapel.  Today we’ll take a closer look at how you can use your opening two buys on Turns 1 and 2 to get to $5 on Turns 3 and 4, so you can play your critical $5 Action before the Turn 5 reshuffle.

Note: these probabilities are all calculated for Turns 3 and 4 together and assume no opponent interference (e.g. Militia or Masquerade).  They also assume you opened with Silver/___; if you opened with some other terminal Action instead of the Silver, you should take into account the 30.3% chance of your opening buys colliding.

Moneylender

We’ll start with Moneylender, to illustrate the probabilities for a generic Silver/Silver or Silver/[Silver equivalent].  It should be clear that the Moneylender probabilities are identical to that of Silver, since it always nets you $2.  Of course, it is better than Silver because it helps trash your Coppers, but it won’t help your buying power any.

At least one $6 or better 42.4%
Not drawing $5 on either turn 8.8%
Drawing $5 or better twice 14.9%

Baron

Baron significantly improves your odds of getting the $6 or $7 (you can even get an $8 with it), but it comes at a cost: a slightly greater chance of not drawing $5 on either turn, when compared to Silver/Silver.  Somewhat paradoxically, it does give better odds of drawing $5 or better twice.

At least one $6 or better 67.7%
Not drawing $5 on either turn 12.6%
Drawing $5 or better twice 27.7%

Feast

Feast is an unusual case in that you are basically guaranteed a $5 card so long as it doesn’t fall in the last two cards of your shuffle.  Moreover, even if it does, you can still often get $5 from your Treasure alone.  Naturally, it greatly reduces the chance you draw $6.

At least one $6 8.8%
Not drawing $5 on either turn 5.0%
Drawing $5 or better twice 29.0%

Quarry

Quarry’s probabilities turn out almost identical to Feast: a slightly lower chance of drawing two $5’s, but you don’t have to trash the Quarry.  Of course, you won’t be able to use a Quarry $5 to gain a non-Action card, but that can be safely ignored.

At least one pure $6 8.8%
Not drawing $5 (for an Action) on either turn 5.0%
Drawing $5 (for an Action) or better twice 27.2%

Horse Traders

As calculated by WanderingWinder in the forum here.  As it turns out, it is very close to Quarry and Feast, but Horse Traders does much better at getting to $6.  Of course, it becomes less effective in the midgame compared to Quarry, when you are discarding better cards.  It is marginally more effective than Silver at getting at least one $5, slightly worse at getting at least one $6, but considerably better at getting multiple $5’s.

At least one $6 or better 38.7%
Not drawing $5 on either turn 5.1%
Drawing $5 or better twice 27.2%

Sea Hag

Sea Hag’s probabilities start off a lot like Feast, except without the free $5, which basically kills your buying power.  In exchange for your monster attack, you get greatly worsened probabilities.

At least one $6 or better 8.8%
Not drawing $5 on either turn 50.6%
Drawing $5 or better twice 0%

Bishop

As far as buying power goes, Bishop is like a Baron that only earns $1.  This puts it well worse than Silver/Silver, but moderately better than Sea Hag.

At least one $6 8.8%
Not drawing $5 on either turn 35.4%
Drawing $5 or better twice 0.8%

Conclusion

Feel free to produce more data for other common openings.  The most glaring omission is Mining Village; my primitive Excel-based simulations are unable to simulate the Mining Village card draw, but I would suspect it would perform as well as Horse Traders in getting to $5 and $6.  Certainly it is strictly superior to Silver.

Bonus Treasure Map Section

We have an excellent writeup of the probabilities of activating your Treasure Maps in the forum by david707.  The summary is: given n cards in your deck, and t Treasure Maps total, the probability of activating them is:

Treasure Map activation probability formula

Put in a more accessible way, this graph illustrates the probabilities as the deck size and number of Treasure Maps increase.  The x-axis represents the number of cards in deck; the y-axis represents probability of activation; and each individual line represents the number of Treasure Maps in deck.

Treasure Map activation probability graph

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29 Responses to Opening Probabilities, Part II

  1. guided says:

    Well I made an attempt at running the math on Mining Village, and the numbers at least seem plausible to me. Please feel free to check my work, particularly listing the cases. I easily could have missed something. I also could have done the permutation counting for any given case wrong😉 When I enumerate cases, C=Copper, S=Silver, M=Mining Village, E=Estate.

    $6+ at least once: 326592000 / 12! = 68.18% [fixed]
    (CCCCS, or M in 5-card hand plus 5 other cards CCCCC, CCCCE, SCCCC, SCCCE, or SCCEE)

    $5 never: 24192000 permutations, 5.05%
    (first 2 hands must be CCCCE + CCCEE or CCSEE + CCCCE)

    $5+ twice: 120960000 permutations, 25.25%
    CCCCC + M/CCSEE
    CCCCC + M/CSEEE
    CCCCS + M/CCCEE
    CCCSE + M/CCCCE
    CCCSE + M/CCCEE

    • david707 says:

      I prefer to use combinations rather than permutations, which usually results in smaller numbers. I’ll check you Math later when I have some time.

      • guided says:

        This is the first time I’ve used permutations for this kind of analysis instead of combinations, following on another commenter’s permutation-based analysis for a much earlier post. I find it’s a bit more of a logical straight line for me to follow since I don’t have to separate ordering effects that matter from ordering effects that don’t matter.

        On checking my work, looks like I grossly miscalculated the “$6+ at least once” probability. It’s actually much higher, around 68%. Which makes sense to me: in my experience MV and Baron are in the same class in terms of getting to $6 before turn 5. Looks like I had dropped a factor of 10 in an intermediate calculation… oops!

        A bit more detail on counting permutations for each case, with “AcB” indicating “number of combinations of B items chosen from A items”:

        CCCCS: 7c4 * 1c1 * 5! * 2 * 7! = 42336000
        (4 out of 7 Coppers) * (1 out of 1 Silvers) * (ways to order these 5 cards) * (could be turn 1 or turn 2) * (ways to order the other 7 cards in the deck)

        M/CCCCC: 1c1 * 7c5 * 5 * 5! * 2 * 6! = 18144000
        (1 out of 1 Mining Villages) * (5 out of 7 Coppers) * (5 positions for the Mining Village since it can’t be the 6th card) * (ways to order the 5 Coppers) * (could be turn 1 or turn 2) * (ways to order the other 6 cards in the deck)

        M/CCCCE: 1c1 * 7c4 * 3c1 * 5 * 5! * 2 * 6! = 90720000
        M/SCCCC: 1c1 * 1c1 * 7c4 * 5 * 5! * 2 * 6! = 30240000
        M/SCCCE: 1c1 * 1c1 * 7c3 * 3c1 * 5 * 5! * 2 * 6! = 90720000
        M/SCCEE: 1c1 * 1c1 * 7c2 * 3c2 * 5 * 5! * 2 * 6! = 54432000

        CCCCE + CCCEE: 7c4 * 3c1 * 5! * 3c3 * 2c2 * 5! * 2! * 2! = 6048000
        (4 out of 7 Coppers) * (1 out of 3 Estates) * (ways to order these 5 cards) * (3 out of 3 remaining Coppers) * (2 out of 2 remaining Estates) * (ways to order these 5 cards) * (ways to order these 2 hands on turn 1 and turn 2) * (ways to order the other 2 cards in the deck)

        CCSEE + CCCCE: 7c2 * 1c1 * 3c2 * 5! * 5c4 * 1c1 * 5! * 2! * 2! = 18144000

        CCCCC + M/CCSEE: 7c5 * 5! * 1c1 * 2c2 * 1c1 * 3c2 * 5 * 5! * 2! * 1! = 9072000
        (5 out of 7 Coppers) * (ways to order the 5 Coppers) * (1 out of 1 Mining Villages) * (2 out of 2 remaining Coppers) * (1 out of 1 Silvers) * (2 out of 3 Estates) * (5 positions for the Mining Village) * (ways to order the other 5 cards in the Mining Village hand) * (ways to order the two hands on turn 1 and turn 2) * (ways to order the remaining 1 card in the deck)

        CCCCC + M/CSEEE: 7c5 * 5! * 1c1 * 2c1 * 1c1 * 3c3 * 5 * 5! * 2! * 1! = 6048000
        CCCCS + M/CCCEE: 7c4 * 1c1 * 5! * 1c1 * 3c3 * 3c2 * 5 * 5! * 2! * 1! = 15120000
        CCCSE + M/CCCCE: 7c3 * 1c1 * 3c1 * 5! * 1c1 * 4c4 * 2c1 * 5 * 5! * 2! * 1! = 30240000
        CCCSE + M/CCCEE: 7c3 * 1c1 * 3c1 * 5! * 1c1 * 4c3 * 2c2 * 5 * 5! * 2! * 1! = 60480000


        $6+ at least once: 326592000 / 12! = 68.18%
        $5 never: 24192000 / 12! = 5.05%
        $5+ twice: 120960000 / 12! = 25.25%

        • guided says:

          Hey, theory, would you mind correcting my initial comment with the wrong $6+ probability and number of permutations?

  2. Anon says:

    Interestingly, Moneylender/Silver ranks above Baron/Silver in councilroom statistics.

    • thisisnotasmile says:

      Moneylender trashes bad cards whereas Baron only discards them and can even gain them in some situations. It should also be noted that “Moneylender probabilities are identical to that of Silver, since it always nets you $2” is not exactly true. Drawing SMEEE on turn 3 or 4 will not make the Moneylender net you $2. However that does not not affect the stats listed as the difference between a $2 hand and a $4 hand is not considered by any of the stats.

      • Death to Sea Hags says:

        If you draw SMEEE then all that’s left is copper. You’re guaranteed $5 on turn 4.

        • thisisnotasmile says:

          Well yeah, but if you open Silver/Silver and draw SSEEE on turn 3, you get a $4 hand and guaranteed a $5 turn 4. However, none of the 3 stats considered in this article differentiate between $2/$5 and $4/$5 turns 3 and 4 so the stats will be the same as a Silver/Silver opening even though there is a chance that your Moneylender will not net you $2.

    • Death to Sea Hags says:

      Although Moneylender is MUCH more likely to hit than Baron – reducing variance – variance reductions in the opening aren’t reflected in win rates. But there may be a difference in expected value.

      Baron/Silver, odds to draw Baron:
      On turn 5 16.6%
      On turn 3,4 83.4%
      —-
      4 Coppers 8.9%
      1 Silver, 3 Coppers 8.8%
      any estates 82,3%

      That 83.4% is fixed for Moneylender on t3,4. Moneylender will give $3 100% of times it appears in t3,4, for a t3,4 expected value of $3. Baron will give $4 only 82.3% of the times it appears in t3,4, for a t3,4 e.v. of $3.29.

      So Baron has a *higher* expected value!

      The difference must be the trashing, as TINAS says, plus your Moneylender is good for more plays in turns 5-10 unless you’re buying extra estates.

      • Anon says:

        In which case I’m still baffled, because if trashing makes up for the difference then I would have expected Baron/Loan to outperform Baron/Silver, but that’s not happening.

      • vidicate says:

        Moneylender will give $3 100% of times it appears in t3,4

        …except with 3 Estates and the Silver, right? I mean, didn’t you just have this discussion above? o.O
        ;-/

  3. Death to Sea Hags says:

    Hags – speaking as one who knows – can really get you stuck under a $4 ceiling if there’s no other good $4 cards.

    t5 Hags + no $5 = death. Sure you can buy another with your $4, but seriously – just give up. This happens 7.6% of the time.

  4. Regarding Treasure Map: I did an extended Treasure Map simulation, pitting various Treasure-Map-buying strategies against TreasureBot. (TreasureBot buys no actions. It always buys Provinces with $8+, or the best treasure card possible, or duchies and estates once there are 4 or fewer Provinces remaining.) If a TMapBot buys two Treasure Maps as soon as possible and then waits for them to trigger, buying no other actions, it does exactly as well as TreasureBot. Seriously, after 10,000 runs, the win percentage was identical (+-0.04%, with within the margin of error) for both 2-player and 4-player games.

    The best TMapBot I found buys up to four, until they trigger. Afterward, extra ones are trashed just to get rid of them. This TMapBot still only beats TreasureBot 52%-42% of the time (with the rest ties). Rather marginal. Having other actions to help set up your Treasure Maps (e.g. Warehouse) might help, but still, I find it’s usually safe to ignore Treasure Map.

    • guided says:

      The best simply-described 2p Big Money strategy I’ve encountered starts buying Duchy instead of Gold when there are 5 Provinces remaining, and starts buying Estate instead of Silver when there are 2 Provinces remaining. You might try that for your TreasureBot instead.

      • Thanks for the note, and I decided to check on it. Further testing has shown the following.

        In a four-player game where everyone is playing as a TreasureBot, the best strategy is to buy duchies with $5-$7 whenever there are 8 or fewer provinces left in the supply, and estates when there are 6 or fewer provinces. That’s earlier than I would have thought! In a three-player game, the most successful TreasureBot (against other TreasureBots) buys duchies when there are 7 or fewer provinces left, and buys estates at 5. And in a 2-player game, the best buys duchies when there are 5 or fewer provinces, and estates when there are 3 or fewer. This does not take into account how large your draw pile is compared to your discard pile, and the stats can be expected to vary in games with actions.

        For a 4-player Prosperity game, the best TreasureBot I found buys provinces with $8-$10 when there are either 10 or fewer provinces in the supply pile or 10 or fewer colonies in the supply pile, whichever is less. It buys duchies (with $5-$7) at the same cut-off: 10. And it bought estates at 6, although buying estates made a negligible difference in win percentage.

        In my analysis, it turns out that there is never a time when you should buy a duchy with $5, but not with $6. And there is never a time when you should buy a province with $8, but not with $9 (in a Prosperity game). In other words, when it’s time to go green, go green all the way.

    • DannyR says:

      “Having other actions to help set up your Treasure Maps (e.g. Warehouse) might help, but…”

      That little line makes all the difference. You’re basic point is valid, that treasure maps by themselves aren’t worth the trouble. But there are several opening cards that make the likelihood of activation skyrocket. Warehouse is one, Haven is another. Thus I find the graph that theory shared quite misleading, because it assumes there is no +card draw anywhere in your deck. Anyone who can build a 30+ card deck with absolutely no drawing power deserves to lose, with or without treasure maps.

      I usually evaluate Treasure Map decks with three assumptions:
      1) that Treasure Maps are the way to go only if there is a cheap drawing card to facilitate them,
      2) that Treasure Maps have to be done immediately, before the deck grows, and
      3) that there’s no point in buying more than 2 Treasure maps. Activate your initial pair, then do more productive things with your gold.

      Obviously, even if these three criteria are met, there are sometimes better strategies that just make Treasure Maps peripheral, or even downright wrong. (Just yesterday my Pirate Ships gorged themselves silly on my opponent’s Treasure Map gold.)

      As far as opening probabilities, I’d be interested to see some statistics on Warehouse/Silver, but even more interested in stats on Treasure Map/Warehouse or Treasure Map/Haven, as examples. Stats on Treasure Maps by themselves are misleading.

      • What’s wrong with a 3rd Treasure Map? It’s self-trashing.

        • DannyR says:

          Self-trashing a 3rd TM spends your action for that turn and cuts you down to a four-card hand. Your 3rd 4-coin purchase could have been spent on a significantly more useful terminal action. Even silver is better than wasting time with a 3rd TM.

          • theory says:

            You can make a plausible case for buying literally nothing instead of the third TM, but buying another terminal or a Silver is asinine. Your only chance of victory once you buy a Treasure Map is to activate it, and you should be willing to go to any lengths to be able to do that. In every simulation we’ve done, the best route to success is to buy as many TM’s as you can so long as they haven’t triggered.

            • As I indicated above, when playing against simple TreasureBots, the Treasure-Map-buying-strategy that worked best in my simulations bought up to four until they triggered. (Buying three, or buying five, had a lower win-percentage; four really was the sweet spot.) Since that TMapBot buys no other actions, using an action to trash a Treasure Map was not much of a downside.

              In a real game where you’re buying other actions afterward, I can’t imagine five would ever work better than four.

              Then again, in those strategies where buying two Treasure Maps works better than buying four, it’s always been the case that buying zero is better than buying two (in my experience).

            • DannyR says:

              (I think) I stand corrected on the terminal action/silver, but I should have said a non-terminal, especially another drawing non-terminal. A second Warehouse, in other words. Yes, the point is still to activate them asap, but I’m just not convinced buying more TMs is the best way to activate them.

              As for the simulations, I’ve never yet heard you say that you worked Warehouses or Havens into those simulations – which is really the only point I’m trying to make. I tried with Geronimoo’s simulator, but its built-in play rules aren’t correct for this situation. For example, it buys 2 TM and 2 Warehouses on turns 1-4, then it (often) plays a Warehouse with 1 TM in hand, drawing the other Warehouse. But since it doesn’t see the second TM, it discards the first one. Then it plays the second Warehouse, drawing the second TM, but since the first TM is already discarded, it discards the second TM too. A real player, on the other hand, would have the wisdom to play both Warehouses, keeping the first TM in hand.

              I’d be happy to see a simulation of the basic TM and Warehouse/Haven starting strategy I’m talking about. Once we have that simulation, we can compare it to buying multiple TMs, etc.

  5. Chris Morrow says:

    For purposes of cards like Explorer and Tournament, which Silver opening has the highest chance of a super-early $8?

    By coincidence, before seeing this new post, I’d done some calculations. Please correct me if I’m wrong, but I find that Baron gives a 7.95% chance of an $8 Turn 3, while with Coppersmith it’s 8.9%. I wasn’t sure how to extend those calculations into Turn 4.

    • DannyR says:

      I haven’t done any calculations, so don’t take my word for it, but it seems to me that coppersmith should give a significantly higher chance. Not only is copper more likely to come up than estate, but also you can get $8 with two possible draws: CS-C-C-C-C or CS-S-C-C-C; whereas with Baron you only have one choice to get $8: B-E-S-C-C.

      • Chris Morrow says:

        Yeah, I took that into account and was suprised it wasn’t much more. I suppose the fact that the CS hands can’t have any estates, a card you have duplicates of, makes some difference.

        The number of ways to get B-E-S-C-C is (3 choose 1) times (7 choose 2); at least, that’s what it is if we consider each Estate and Copper to have a unique identity, which shouldn’t make any difference for the calculations’ result. This total is 63. Meanwhile, the number of ways to get CS-C-C-C-C is (7 choose 4), and the number of ways to get CS-S-C-C-C is (7 choose 3). Each of these is 35, so that total is 70 — only 7 more than the Baron one.

        The total number of possible hands in all these cases is (12 choose 5), or 792. So the Baron’s fraction is .079(54), and the Coppersmith’s is .088(38). It looks like I accidentally rounded that one up, not that it makes an enormous difference. Yay math!

        • guided says:

          If you want the total probability for turn 3+4, it’s just doubled. 17.7% for Coppersmith, 15.9% for Baron.

          • Anonymous says:

            No it’s not. These aren’t independent events so adding the probabilities does not give the probability of an “either or”. In other words, drawing or not drawing an $8 on turn 3 changes the odds that you will/won’t draw an 8 on turn four. If the draws of turn 3 and turn 4 were not related then your numbers would be correct. But they are both quite related, in that they draw from the same card pool.

            • WanderingWinder says:

              And yet it is, since you don’t know anything about the distribution of cards beforehand. Now, the probability that you get it on turn 4 HAVING ALREADY SEEN turn 3 is quite different, but I don’t think that’s what he was talking about.

  6. Anonymous says:

    Moneylender/Silver is not quite the same as Silver/Silver. If you get Moneylender with 3 Estates and a Silver, that negates your Moneylender for that run through the deck. If that Moneylender was a Silver, you would get $4 on that turn rather than $2. This is of course a rare case, but I think it is worth mentioning.

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