Crypt Drawing Probabilities

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Hey All,

I have been playing quite a few "star" vampire decks lately and wanted
to do some number crunching. I think these numbers are pretty darn
close (maybe have some rounding errors), but I welcome anyone to check
my work.

This mainly came from conventional wisdom saying that 4 vampires in 12
is how many you want if you want to see your chosen vampire, but 5 in
12 is what you should use if you *need* your chosen vampire. As such, I
have been often using 4/13 or 5/13 in certain decks because I think
there must be a better way. I finally got around to crunching the
numbers and thought I'd share.

Jeff

===========================

CRYPT DRAW PROBABILITIES

Here's the formula you need:

% = 100(n choose m)(12-n choose 4-m)/(12 choose 4)

where 'n' is the number of copies of the vampire in question and 'm'
is the number of copies of that vampire that you're looking for the
odds on. (Note that this presumes that you have a crypt of 12 and are
initially drawing 4. Otherwise you'll need to modify the 12 and the
4.) I'm using (x choose y) to represent:

(x choose y) = x!/((x-y)!y!)

which is the standard form for the number of distinct combinations of
y in x.



TWELVE (12) VAMPIRE CRYPT
# of copies \ % odds of drawing
0 1 2 3 4
0 100.0 0 0 0 0
1 66.7 33.3 0 0 0
2 42.4 48.5 9.1 0 0
3 25.5 50.9 21.8 1.8 0
4 14.1 45.3 33.9 6.5 0.2
5 7.1 35.4 42.4 14.1 1.0
6 3.0 24.2 45.5 24.2 3.0



THIRTEEN (13) VAMPIRE CRYPT
# of copies \ % odds of drawing
0 1 2 3 4
0 100.0 0 0 0 0
1 69.2 30.8 0 0 0
2 46.2 46.2 7.7 0 0
3 29.4 50.3 18.9 1.4 0
4 17.7 47.0 30.2 5.0 0.1
5 9.7 39.2 39.2 11.2 0.7
6 4.8 29.4 44.1 19.6 2.1



COMBINED TABLE FOR POPULAR COMBINATIONS (#/crypt size)
# of copies \ % odds of drawing
0 1 2 3 4 1 or 2
3/12 25.5 50.9 21.8 1.8 0 72.7
4/13 17.7 47.0 30.2 5.0 0.1 77.2
4/12 14.1 45.3 33.9 6.5 0.2 79.2
5/13 9.7 39.2 39.2 11.2 0.7 78.4
5/12 7.1 35.4 42.4 14.1 1.0 77.8



LIKELIHOOD OF SEEING CHOSEN VAMPIRE EACH OF SEVERAL GAMES
Distribution \ Non-Zero Chance in 1/3/4 Games
1G 3G 4G
3/12 74.5 41.3 30.8
4/13 82.3 55.7 45.9
4/12 85.9 63.4 54.4
5/13 91.3 73.6 66.5
5/12 92.9 80.2 74.5



COMMENTARY
With 3/12 distribution:
* You are more than 50% likely to see your chosen vampire in any game.

With 4/13 distribution:
* You are more likely to see 2 of your chosen vampire than zero in any
game.
* You are more than 50% likely to see your chosen vampire
in all rounds of a 2+F tournament.

With 4/12 distribution:
* You are nearly 2/3 likely to see your chosen vampire
in all rounds of a 2+F tournament.
* You are more than 50% likely to see your chosen vampire
in all rounds of a 3+F tournament.

With 5/13 distribution:
* You are more likely to see 3 of your chosen vampire than 0 in any
game.
* You are nearly 3/4 likely to see your chosen vampire
in all rounds of a 2+F tournament.
* You are nearly 2/3 likely to see your chosen vampire
in all rounds of a 3+F tournament.

With 5/12 distribution:
* You are over 4/5 likely to see your chosen vampire
in all rounds of a 2+F tournament.
* You are nearly 3/4 likely to see your chosen vampire
in all rounds of a 3+F tournament.

 

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jeff_kuta wrote:
| Hey All,
|
| I have been playing quite a few "star" vampire decks lately and wanted
| to do some number crunching. I think these numbers are pretty darn
| close (maybe have some rounding errors), but I welcome anyone to check
| my work.

The methodology is fine; rounding errors will produce at most .1% error,
which just isn't enough to throw stones at.

If there are mistakes, we all made the same ones.

| This mainly came from conventional wisdom saying that 4 vampires in 12
| is how many you want if you want to see your chosen vampire, but 5 in
| 12 is what you should use if you *need* your chosen vampire. As such, I

....Alex Broadhead and I worked these numbers out some time back and
posted the results here. That's pretty much where all the "conventional
wisdom" came from, I think. 😉

| have been often using 4/13 or 5/13 in certain decks because I think
| there must be a better way. I finally got around to crunching the
| numbers and thought I'd share.

Ah, optimism. 😉

| COMBINED TABLE FOR POPULAR COMBINATIONS (#/crypt size)
| # of copies \ % odds of drawing
| 0 1 2 3 4 1 or 2
| 3/12 25.5 50.9 21.8 1.8 0 72.7
| 4/13 17.7 47.0 30.2 5.0 0.1 77.2
| 4/12 14.1 45.3 33.9 6.5 0.2 79.2
| 5/13 9.7 39.2 39.2 11.2 0.7 78.4
| 5/12 7.1 35.4 42.4 14.1 1.0 77.8

Looking only at the 5/13 and 5/12 scenarios, here we see a more even
distribution towards the "1" or "2" chances by increasing the crypt
size... at the cost of a sharp increase of not seeing the vamp at all.

(Compare 78.4 - 77.8 = .6% to 9.7 - 7.1 = 2.8%.)

However, it DOES indicate that if you're going to have a semi-star vamp
deck (for example, my Adonai + Omaya deck, which can limp along with
either while it chases the second one, has 3 copies of each in it)... it
might be wiser to back off a little bit of the Main Guy so as to be sure
you don't crypt-screw yourself, either by decreasing # of copies or
increasing crypt size.

The 3/12 odds in the A&O deck give me a 75% chance of getting either,
which means I get the 95 percent chance of at least one (6 copies,
above)... But at the same time, it's not likely that I'll ever get all 3
of one guy (worst draw ever), and it's very unlikely that I'll get 2 of
each (second worst draw ever) -- so I should always be able to bring out
3 vamps without fishing. Which is critical for that deck.

Similarly, 5/13 might do well for a "Beast + Weenies" POT deck; and
while I know David Davila has an amusing story to tell about THIS, it's
worth noting that getting 3+ Beast is not really what you want, because
you NEED the weenie support for him, and the deck can typically limp
along until you get Beast.

However, for the traditional Fatima + AR deck, you GOTTA have Fatima.
That means 5/12 the whole way, no questions asked; you can take the time
to fish for weenies, but without Fatima you're defenseless.

| LIKELIHOOD OF SEEING CHOSEN VAMPIRE EACH OF SEVERAL GAMES
| Distribution \ Non-Zero Chance in 1/3/4 Games
| 1G 3G 4G
| 3/12 74.5 41.3 30.8
| 4/13 82.3 55.7 45.9
| 4/12 85.9 63.4 54.4
| 5/13 91.3 73.6 66.5
| 5/12 92.9 80.2 74.5

Amazing what a percent and a half will add up to over time, isn't it?

Even the difference between 86% and 93% is worth commenting on; that's a
lot bigger than it looks when you have to have it IMMEDIATELY.

| With 5/13 distribution:
| * You are more likely to see 3 of your chosen vampire than 0 in any
| game.
| * You are nearly 3/4 likely to see your chosen vampire
| in all rounds of a 2+F tournament.
| * You are nearly 2/3 likely to see your chosen vampire
| in all rounds of a 3+F tournament.
|
| With 5/12 distribution:
| * You are over 4/5 likely to see your chosen vampire
| in all rounds of a 2+F tournament.
| * You are nearly 3/4 likely to see your chosen vampire
| in all rounds of a 3+F tournament.

....leading to what we finally concluded, which is that:

- -- Increasing crypt size by ANY amount typically only increases variance
somewhat, while reducing your overall chance of seeing the Star Player.

- -- The difference between 5 copies and 4 copies is huge. It's always
easier to fish for weenies than it is to fish for Fatima.

I don't have time to go back and fish out the numbers right now, but
often, "fishing once" is something every deck has time for -- if you
find that your deck CAN do this, then when deckbuilding, you can run the
numbers for 5 cards drawn instead of 4, and consider the ratios that way.

- --
Derek

insert clever quotation here

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Archived from groups: rec.games.trading-cards.jyhad (More info?)

Howdy,

> | This mainly came from conventional wisdom saying that 4 vampires in
12
> | is how many you want if you want to see your chosen vampire, but 5
in
> | 12 is what you should use if you *need* your chosen vampire. As
such, I
>
> ...Alex Broadhead and I worked these numbers out some time back and
> posted the results here. That's pretty much where all the
"conventional
> wisdom" came from, I think. 😉

Speak of the devil... Here's my favorite charts:

Crypt: Assumes 12 vampires, initial draw of 4

+ % odds of drawing
# of copies 0 1 2 3 4 at least 1
1 66.67 33.33 0.00 33.33
2 42.42 48.48 9.09 0.00 57.58
3 25.45 50.91 21.82 1.82 0.00 74.55
4 14.14 45.25 33.94 6.46 0.20 85.86
5 7.07 35.35 42.42 14.14 1.01 92.93
6 3.03 24.24 45.45 24.24 3.03 96.97
7 1.01 14.14 42.42 35.35 7.07 98.99
8 0.20 6.46 33.94 45.25 14.14 99.80
9 0.00 1.82 21.82 50.91 25.45 100.00
10 0.00 9.09 48.48 42.42
11 0.00 33.33 66.67


Crypt: Assumes 12 vampires, drawing 5

+ % odds of drawing
# of copies 0 1 2 3 4 at least 1
1 58.33 41.67 0.00 41.67
2 31.82 53.03 15.15 0.00 68.18
3 15.91 47.73 31.82 4.55 0.00 84.09
4 7.07 35.35 42.42 14.14 0.20 92.93
5 2.65 22.10 44.19 26.52 4.42 97.35
6 0.76 11.36 37.88 37.88 11.36 99.24
7 0.13 4.42 26.52 44.19 22.10 99.87
8 0.00 1.01 14.14 42.42 35.35 100.00
9 0.00 4.55 31.82 47.73
10 0.00 15.15 53.03
11 0.00 41.67


Crypt: Assumes 12 vampires, drawing 6

+ % odds of drawing
# of copies 0 1 2 3 4 at least 1
1 50.00 50.00 0.00 50.00
2 22.73 54.55 22.73 0.00 77.27
3 9.09 40.91 40.91 9.09 0.00 90.91
4 3.03 24.24 45.45 24.24 3.03 96.97
5 0.76 11.36 37.88 37.88 11.36 99.24
6 0.11 3.90 24.35 43.29 24.35 99.89
7 0.00 0.76 11.36 37.88 37.88 100.00
8 0.00 3.03 24.24 45.45
9 0.00 9.09 40.91
10 0.00 22.73
11 0.00


As Derek says, many decks can get away with fishing once, so the 'draw
5' chart is useful. The 'draw 6' chart at bottom is much more
fanciful, and I figured any higher draw numbers were pretty much
useless, though I suppose that a deck that plays a lot of Effective
Management or Kindred Intelligence or equivalent might care. I didn't
bother with crypts larger than 12, as I rarely/never use them.

It is important to note that the body of the tables is the % chance of
drawing _exactly_ that number of the vampire; that's why the numbers
peak in the middle for most rows/columns. The last column is the
chance of drawing at least one, i.e. one or more, which is more
important to most decks than the 'exactly' number.

The breaks in the 4 chart are 3 (51% chance of 1, 75% chance of at
least 1), 4 (86% chance of at least 1), and 5 (93% chance of at least
1, with rapidly diminishing returns with more crypt copies).

The breaks in the 5 chart are, as expected, lower, with 2 (53% chance
of 1, 68% chance of at least 1), 3 (84% chance of at least 1), and 4
(93% chance of at least 1, diminishing returns) roughly equating to the
effects of one more copy on the 4 chart (though the 68% chance of at
least 1 with 2 copies is actually a much better statistic, as it means
less potential duplication).

Anyway, hope that helps,
Alex

 

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wumpus wrote:

> As Derek says, many decks can get away with fishing once, so the
'draw
> 5' chart is useful. The 'draw 6' chart at bottom is much more
> fanciful, and I figured any higher draw numbers were pretty much
> useless, though I suppose that a deck that plays a lot of Effective
> Management or Kindred Intelligence or equivalent might care. I
didn't
> bother with crypts larger than 12, as I rarely/never use them.
>

I was playing Tyler Archon Multi-Rush in a tournament when, at the
time, you could rush as many times as you wanted with an Archon on
Tyler. It had 5 copies of Tyler in a 12 card crypt. I did the math once
to find out the chance of all 5 Tylers being on the bottom of the
crypt. Its a really small percent. Well, it happened. I had a Tzi prey
who didn't go backward, just wall up and a !Salubri predator that was
mostly busy with its predator. I got out some of my support vampires
and fished all the way until it had to be Tyler. I was still able to
influence Tyler out. Unfortunately, the Tzi wasn't ever going to let me
get off a +1 Stealth Vast Wealth action to get a Ring of Rowan or call
an Archon on Tyler. I lost and I think the Tzi won the table, oh well.
Statistically, I don't think I can play enough games of Vtes for that
to happen again. Too bad it was at a tournament though.

Later,
~Rehlow
--who still believes 5/12 is the way to go for star vamp decks

 

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Try my Fragment of Elder Library Deck Builder program.

In the "Draw Crypt" you can show the probabilities.
If anyone has an idea what I'd show more, please tell me.
regards,
Bala

ps: web.interware.hu/bala0/feldb.htm

 

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Archived from groups: rec.games.trading-cards.jyhad (More info?)

Maybe I can put this method to my program.

I show only the summa capacity of the 4 vampires in uncontrolled
region.
I think I'll put this data to: which vampires - which probality will be
in your unctrolled region.