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MrBenn wrote:I've spent a but of time dusting off some textbooks, and have put together a quick spreadsheet that will calculate the probability of dropping a particular bonus region on a map, that also takes 'build-your-own' bonuses into consideration...
Bonus Probabilities Spreadsheet v1
I don't think I've got my formulae wrong, but it has been a few years since I did any meaningful work using probabilities like this... If people think it would be useful, I can expand the spreadsheet to allow for more regions to be calculated simultaneously?
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john9blue wrote:My preliminary test didn't work, though. Try 9 territories with a continent of 3... each player gets 3 territs in a 2p or 3p game, and 84 ways to distribute them, so the odds are 1/84, or 1.19%. Your spreadsheet says 1.23%, so you're gonna have to tweak it.
Merciless Wong wrote:the.killing.44 wrote:Merciless Wong wrote:The issue is not just the cities. Its the Cities and the Challenges. The Cities is a 9% chance of giving someone an edge.
The Challenges, another 9% or so (2/27). So your chance of having no starting bonuses or so is about (1-9%)^2.
So you are talking about a 18% chance of either 1 or 2 starting with an advantage - which would beat the 10% level you quoted
I'm not going to go into your other complaint, but your math is wrong. If you're saying that it's 9^2 with a max of 18, 9^2 = 81. Which is blatantly wrong.
.44
I'm not sure if you noticed but ^ is the power symbol.
(1-9%)
=91%
91%^2
=approx 81%
Chance of no bonus is 81%
Chance of bonus is 100% -81% = 19%
.
Here's my working:
2^5 = 32. There are 5 provinces with cities, assuming conquer club will place them all with on player or the other there are
2x2x2x2x2 possibilities.
In 1 of them player 1 has it all
In 1 of them player 2 has it all
In 5 of them player 1 has 4 of the 5 cities
In 5 of them player 2 has 4 of the 5 cities
That's my 12 out of 32 = 37%
The reamaining 20 out of 32 are scenarios where player 1 has 3 cities (10 possibilities)
and player 2 has 3 cities (10 possibilities)
If you wish to model the number of neutals in 1v1 (I presume its 1 out of 40+ territories) it should make a slight difference.
If its 3 starts (1 neutral, meaning neutals will be 1/3 of the board):
3^5= 243 scenarios (focuing only on non hard coded neutral provinces that matter for city bonus)
2 of them they have it all
5 of them p 1 has 4, neutrals have 1
5 of them p 2 has 4, neutrals have 1
5 of them p 1 has 4, other has 1
5 of them p 2 has 4, other has 1
Makes it 22 out of 243 = 9.1%
the.killing.44 wrote:
Merciless Wong wrote:
The issue is not just the cities. Its the Cities and the Challenges. The Cities is a 9% chance of giving someone an edge.
The Challenges, another 9% or so (2/27). So your chance of having no starting bonuses or so is about (1-9%)^2.
So you are talking about a 18% chance of either 1 or 2 starting with an advantage - which would beat the 10% level you quoted
I'm not going to go into your other complaint, but your math is wrong. If you're saying that it's 9^2 with a max of 18, 9^2 = 81. Which is blatantly wrong.
.44
I'm not sure if you noticed but ^ is the power symbol.
(1-9%)
=91%
91%^2
=approx 81%
Chance of no bonus is 81%
Chance of bonus is 100% -81% = 19%
AndyDufresne wrote:**Scratches his head** I'm glad I'm surrounded by you math folks. I'm a word man, always have been.
I remember when the only math we had to figure into cartography was counting territories, counting borders, counting assault routes, and counting bonuses!
Yowza.
--Andy
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--Andy
sailorseal wrote:My big boy banana was out the whole time
AndyDufresne wrote:Forever linked at the hip's-banana! (That sounds strange, don't quote me.)AndyDufresne wrote:Many Happy Bananas to everyone, lets party...with Bananas.
--Andy
MrBenn wrote:OK, here's the rough guide to bonus probabilities... It helps to think of the map a bit like a pack of cards... the total number of territories on the map represents the size of the deck. The territories a player gets is their hand - the goal is to work out the probabiltity of being dealt a set...
The size of the map / deck: T
The number of players: p
Number of territories in each hand: h
h = Floor(T/p,1) *If p=2, then h = Floor(T/3)
From this we can work out the total number of possible combinations of hands:
Combin(T,h) = T!/h!(T-h)! = Fact(T)/Fact(h)-Fact(T-h)
For Classic, with 2 or 3 players, this gives: Combin(42,14) = 52,860,229,080 possible hands.
Of those hands, how many of them will have the Australia bonus?
The total area of Australia: A
Territories required for bonus: a
There is only one way you can hold all four of the four Australian 'cards': Combin(A, a)= Combin(4,4) = 1
the other 10 cards in your hand can be any of the other 38 terrs: Combin((T-a), (h-a)) = Combin (38, 10) = 472,733,756
Multiplying these together will give the total number of ways you can hold the Australia bonus: 1 x 472,733,756 = 472,733,756
Divide this by the total number of combinations for the hand, and you get: 472,733,756/52,860,229,080 = 0.89%
Combin(A, a) x Combin((T-a), (h-a)) / Combin(T,h)
This is a similar method to that you would use to calculate the probability of getting a full house in a game of Poker. The problems come when you start trying to take into account of the build-your-own bonuses - it's more complicated than I originally thought, and involves Hypergeometric Probabilities rather than Binomials...
MrBenn wrote:A nice extension (which I don't intend to develop in the immediate future) would be something that calculated the probability of dropping ANY bonus, and the average/typical bonus dropped.
Merciless Wong wrote:-Note that you aren't doing compounding. The second bonus is identical to the first.
MrBenn wrote:MrBenn wrote:I've updated my Bonus Probabilities Spreadsheet to account for starting positions, but am having trouble uploading it anywhere
Got it.... http://www.fileden.com/files/2009/1/9/2259283//BonusProbability2.xls
Play around with the bonus spreadsheet a bitand see if there is some combination of neutral starts, or larger starting position groups that help. It might be that by ADDING terrs to the starting group, you can increase the number of dropped territories to 19 or 20 for 1v1 games (which is better than 18)
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