I figured out that it’s

**very important to hit $7 after at most two shuffles**. The gaining of coppers has the downside that your first shuffle will take T3-T5, and your next shuffle T6-T9. If by then you did not get any $7-hand, you will have to wait for a counting house in, say, T10-T14; and if your $7 shows up at e.g. T12, you will probably need till T16 to get to five provinces. This is ok on a weak board, but too late compared to a good engine. In other terms:

The odds to hit $7 before a third shuffle is crucial, and to start with,

**I will approximate the simple scenario of opening 3/4 gaining 5 coppers** in total:

In that case, you have 15 cards including 12 coppers and three estates, distributed on three turns, namely T3, T4, T5.

a) The risk to hit $4 three times is 10/14 * 5/13 (approximately 27.5 percent).

b) The chance to hit $5 twice (buying CH twice and two coppers) is 4/14 * 3/13 (approximately 6.6%)

c) in all other cases you get exactly one CH and it is recommended to gain 4 additional coppers (if you buy 5, it makes your decksize 21 cards, that creates an unnecessary 1/21 risk to miss CH next shuffle). The odds of c) are 1-27.5%-6.6% = 65.9%

-> a) If you miss a CH on T3-T5, you won’t get 7 before the next shuffle.

-> b) If you have 2 CH’s, you have 19 cards including 3 estates and 2 CH’s. This leads to three turns T6, T7, T8, and four cards will be seen after a reshuffle in T9 (which is bad). That means, you will need a third shuffle, if both CH’s show up in T6 or T9, the odds are 9/19 * 8/18 =4/19 (approx. 21.1. %).

-> c) You need a third shuffle, if your counting house comes in T6 (25%)

Total: The odds to need three shuffles in (27.5%*100%) + (6.6%*21.1%) + (65.9%*25%) = 45.4%.

The other way round,

** you will be able to hit $7 and rush the provinces before the third shuffle in 55.6%** of all solitaire games starting 4/3.

This is, by now, not the whole story, e.g. you might start 5/2 or topdeck coppers on 4/3. But I think this analysis allows some rough ...

... conclusions:

Basically, it’s a kind of

**coinflip** whether you will get to 5 provinces extremely fast (let’s call this scenario „T12") or solely with a mediocre speed (called „T16").

If there is a strong alternative, that will certainly beat T16 and still has some potential to beat you at T12, the combo is too weak.

If the best alternative will certainly lose against T12 and still have trouble beating T16, the combo is strong.

In middling scenarios, T12 will likely win, and T16 will likely lose. In that case, Cutpurse (or similar) will make the difference, because with the attack, T16 becomes more likely than T12. And so:

- On a weak board, go for it.
- On a mediocre board, go for it if there is no attack like cutpurse.
- Otherwise, skip it if there is an engine/combo that might beat 5 provs by T12.

In other terms:

**Counting House/Travelling Fair is more often the best way to go than not. ** Thx for posting it, gamesou