[GB] Check My Math

[Eliahad]

I was poking around with some numbers, and I need someone to make sure this is right.

First let's go with opening hands for one god.

24! (Total number of cards in the deck) / ((4! (We don't care about order) * 20! (Cards left over after picking)

So:

24*23*22*21*20! / 4*3*2*1*20!

24*23*22*21 / 4*3*2*1

255,024/24

There are 10,626 possible starting hands. Okay! That's a little less random than we used to be great!

So then I decided to check, How many unique occurrences of one game's worth of cards would be without caring about the order those cards are drawn.

24! (Total number of cards in the deck) / ((15! (We don't care about order) * 9! (Cards left over after picking)

24*23*22*21*20*19*18*17*16 / 9*8*7*6*5*4*3*2*1

474,467,051,520 / 362,880

There are 1,307,504 possible draws of the cards to form, essentially a hand of 15 cards. Holy crap!

So then I thought, okay, how many unique draws are there, where you form a hand of four cards and then draw one card at a time, so order does matter.

10,626 (starting hand) * 20*19*18*17*16*15*14*13*12*10*11 (Cards left in the deck each round).

So I get 71,241,227,785,728,000 unique games of Godball...for one god.

Am I doing that wrong? Is my math crazy off? I imagine my math is wrong. Someone tell me what I'm doing wrong?

[Mike]

It checks out. For a 24 card deck, initial draw of 4, pulling 1 new card each round, there's around 71 quadrillion possible combinations.

And I solved this differently than you did (different logic, but ultimately the same calculations) and I got the same answers.

[FlameBlade]

>>> def choose(n,k):
return(math.factorial(n)/(math.factorial(n-k)*math.factorial(k)))

>>> #starting deck.
>>> choose(24,4)
10626.0
>>> choose(24,15)
1307504.0
>>> choose(24,4)*(math.factorial(20)/math.factorial(9))
7.1241227785728e+16


Checks out.

[FlameBlade]

>>> for i in range(2,7):
print(i, (choose(24,4)*(math.factorial(20)/math.factorial(9)))**i)


2 5.075312536417983e+33
3 3.6157149649071447e+50
4 2.5758797342321542e+67
5 1.8350883489507341e+84
6 1.3073394707453475e+101

That's just from card draws. Doesn't count variations in movement on the board!

[Eliahad]

That is so many combinations. The feel of balance will be more important than actual balance, I think.

I think I figured out how to use Flame's hyper-geometry calculator to figure out the chances that the same number of cards show up in 2 games.

In game one you draw a set of 15 cards.

What then, are the chances that you draw the same set of 15 cards, even if it's a different order?

Well, less than the calculator shows

But if I add up the probability of each successive column:

15, 14, 13, etc. I get the probality of different overlaps of cards from each set of the 2 games.

[Eliahad]

No, no, I think I did that wrong... I'll get there.

[Mike]

Type 24 choose 15 into Google.

1,307,504

[Eliahad]

Yes! Now I want to find the probability that, in 2 games, 1 card is different... Or 2 cards... All the way up to the 9 unused cards are different.

[Eliahad]

The odds that 0 cards are different would be: very close to zero percent, but not zero percent.

The odds that at most 1 card would be different: 0.01%

The odds at most 2 cards are different: 0.3%

The odds that at most 3 cards are different: 3.2%

At most 4 cards: 16.4%

At most 5 cards: 45.3%

At most 6 cards: 72.5%

At most 7 cards: 90.2%

At most 8 cards: 94.4%

At most 9 cards are different: 94.8%

At least 9 cards are different: 0.3%

At least 8 cards are different: 4.8%

At least 7 cards are different: 22.5%

At least 6 cards are different: 64.6%

At least 5 cards are different: 93.5%


I've run out of time.