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.
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 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 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’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%|
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’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%|
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%|
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:
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.