[Excel] Bonus Probabilities

[LED ZEPPELINER]

**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

LOL, same. I look at the page, see a bunch of Letters, parenthesese, %'s, *'s, and numbers, and i am like, i'm getting out of here

[MrBenn]

That's not even half of it!

The calculations I showed above were for the probability of ME dropping Australia in a 2 or 3 player game (.89%)

If we take other players into account, then we get the following:
In a 1v1 on Classic, then there is a 1.76% chance of somebody/anybody dropping Australia
In a 3-player game, there is a 2.68% chance of somebody/anybody dropping Australia

I'll try and put these calculations into something user-friendly. I've got it completely figured out for fixed bonuses; but still have a bit of working out to do for build-your-owns

[LED ZEPPELINER]

[oaktown]

For the past hour MrBenn has been schooling me on the mathematics of probability that I should have learned 20 years ago, but you see there was this girl with really long black hair sitting in front of me in math class...

Anyway, the fixed bonuses finally make sense to me.

On flexible bonuses I am close, I think.

[Merciless Wong]

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...

I'm not devoting time to checking this but at the very least, you've only done it for one hand or 1 player. Not 2-3 players. Trust me and simulate or approximate. It will hurt less.

[MrBenn]

Here it is then - and it is finally accurate:

MrBenn's Bonus Probability Calculator

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.

Happy calculating

[AndyDufresne]

It's colorful, like a rainbow. I like rainbows, and ducks, and daisies. Oh, look I see a butterfly.

**Wanders off like a monkey without an understanding of math.**

Good work, though, I think, maybe?


--Andy

[Merciless Wong]

Nice.

-Note that you aren't doing compounding. The second bonus is identical to the first.

-Rounding of number of tertis is also an approximation.

But it should be close enough for discussion.

This stuff should be used for benchmarking anyway. E.g. the chance is less than or more than an Australia start in classic risk using the same model

[MrBenn]

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.

-Note that you aren't doing compounding. The second bonus is identical to the first.

You're right. Each of the probabilities is independent. It got too complicated with the number of permutations with the build-your owns.

[MrBenn]

Subject: Re: Thyseneal: V 3.6 with XML v1.3 [Gp][Gr][Xml] [Beta]

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)