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Brass: Lancashire» Forums » Strategy

Subject: running the numbers on draws from Distant Market rss

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Brandon Irvine
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Massachusetts
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Howdy. This might be old news (and sort of obvious for people who've played a lot):

I wrote up a script to look at all 831600 ways that the distant market tokens can be arranged (in the deck of 12 that they are put on the board). On average, the Distant Market will hit 0 after ~4.48 distant market tokens are revealed.

If you want some other stats, ask in the comments and I'll see if I can whip them up.
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Babis Tsimoris
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Thessaloniki
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Cool! Nice try of yours! cool
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Maarten D. de Jong
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What is the unbiased sample variance? Or, better yet, can you plot the PDF for this distribution? (I'm curious what it looks like.)
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Brandon Irvine
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cymric wrote:
What is the unbiased sample variance? Or, better yet, can you plot the PDF for this distribution?


I don't think a probably distribution function is really what we're looking for here, since there are only 5 possibilities (3 draws up to to 7 draws, because it's mathematically impossible for any more or less, apparently, given the composition of the distant market tiles deck). Anyway, here are the likelihoods for each:

3 draws required: 18%
4 draws required: 35%
5 draws required: 30%
6 draws required: 14%
7 draws required: 03%

In case you care, the standard deviation is about 1.03.

(I rounded on the percentages, but that's probably more info than you need, anyway.)

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Maarten D. de Jong
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I'm actually thinking—but it's just before bed time so I may be talking nonsense—that these numbers aren't what we want to know in the first place. We want conditional probabilities, so for example: Given that the first draw was x, how many draws can we expect to happen still? And given that the second draw was y, how many more can we expect? Of course all those draws average up to your calculated value of ~4.5, but that discards a lot of potentially useful information because we know what values still remain in the draw stack. Of course, these results can no longer be presented in simple tables meeple.

That said, I doubt there are many people who play Brass so well that the above makes a noticable difference on the game's outcome.
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Brandon Irvine
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By the way, I don't think it was clear above when I said "until 0". What I mean is, until the demand track hits "no demand" and cotton mills aren't flipped.

cymric wrote:
Given that the first draw was x, how many draws can we expect to happen still?


Below, the first line means, if you turn over -4 as the first tile revealed, the likelihood is 49% that by the 3rd tile revealed, the Distant Market track will be at "no demand", i.e. where you don't get to flip your cotton mill and no more tiles are revealed. (Rather than "0", where you still flip cotton mills but with 0 added income bonus.)

"3 reveals until no demand" includes the first reveal here. So "3 reveals until no demand" means "2 *more* reveals until no demand".

Average draw where zero is hit: 4.47965367965
Standard deviation: 1.02913062182
If -4 is revealed first...
... 3 reveals until no demand: 0.49
... 4 reveals until no demand: 0.35
... 5 reveals until no demand: 0.14
... 6 reveals until no demand: 0.03
... 7 reveals until no demand: 0.00
If -3 is revealed first...
... 3 reveals until no demand: 0.31
... 4 reveals until no demand: 0.38
... 5 reveals until no demand: 0.23
... 6 reveals until no demand: 0.07
... 7 reveals until no demand: 0.01
If -2 is revealed first...
... 3 reveals until no demand: 0.16
... 4 reveals until no demand: 0.38
... 5 reveals until no demand: 0.30
... 6 reveals until no demand: 0.13
... 7 reveals until no demand: 0.03
If -1 is revealed first...
... 3 reveals until no demand: 0.05
... 4 reveals until no demand: 0.36
... 5 reveals until no demand: 0.35
... 6 reveals until no demand: 0.19
... 7 reveals until no demand: 0.05
If 0 is revealed first...
... 3 reveals until no demand: 0.00
... 4 reveals until no demand: 0.24
... 5 reveals until no demand: 0.44
... 6 reveals until no demand: 0.26
... 7 reveals until no demand: 0.06
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Maarten D. de Jong
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Colour me impressed that you worked out a first set of numbers even though it's just a few more lines of program code. Have a thumb for your efforts. I like how the PDFs change based on the first reveal. It seems you can base a few easy rules of thumb on these results already.
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Len
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I always stop shipping distant after three tokens are revealed. This shows I was a little gun shy.

Very interesting!
 
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it depends on what you draw. If the first 3 tokens are 0,0, -1 then why not try the 4th one?
 
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