## [Excel] Bonus Probabilities

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### Re: Bonus Probabilities

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

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
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### Re: Bonus Probabilities

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

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### Re: Bonus Probabilities

sailorseal wrote:My big boy banana was out the whole time
AndyDufresne wrote:
AndyDufresne wrote:Many Happy Bananas to everyone, lets party...with Bananas.
--Andy
Forever linked at the hip's-banana! (That sounds strange, don't quote me.)
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### Re: Bonus Probabilities

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.

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### Re: Bonus Probabilities

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...

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.
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### Re: Bonus Probabilities

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

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### Re: Bonus Probabilities

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

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### Re: Bonus Probabilities

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
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### Re: Bonus Probabilities

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.

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

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### Re: Bonus Probabilities

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

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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### Re: [Excel] Bonus Probabilities

i have to ask...does this table allow for consideration of coded neutrals in a region when calculating the drop probability?

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### Re: [Excel] Bonus Probabilities

cairnswk wrote:i have to ask...does this table allow for consideration of coded neutrals in a region when calculating the drop probability?

Not sure I follow. A coded neutral means the probability of dropping the bonus is 0.
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### Re: [Excel] Bonus Probabilities

koontz1973 wrote:
cairnswk wrote:i have to ask...does this table allow for consideration of coded neutrals in a region when calculating the drop probability?

Not sure I follow. A coded neutral means the probability of dropping the bonus is 0.

Of course that does, but...
in a 4 tert region...where one of the terts is coded neutral, you still have a drop of 3 to occur, but actually 4 to gain the bonus...so there must be differing probabilities, yes?

I think this tool could be incorporated with the Starting Drop Calculator, so that all that information is on one page.

The info for each two lines is also a little confusing.
Bonus Area Size 4
Terrs Required for Bonus 4

What does bonus area size actually mean?
Terrs Required for Bonus is self explanatory.

Take this for example.
Bonus Area Size 4 Probablity of A player receiving the bonus 1.01%
Terrs Required for Bonus 4 Probablity of ANY player receiving the bonus 2.02%

What does 1.01% mean in relation to Bonus Area Size given that is not properly explained? Or at least i don't understand it, and if i don't understand it, then there are going to be others out there who don't understand it also.
Does 2.02% take into account that perhaps 2 of those terts starts neutral?

cairnswk

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### Re: [Excel] Bonus Probabilities

I'll get in touch with MrBenn soon to give you a more detailed reply and maybe see if he is willing to update the tool (12players etc etc) and join it with the one i did.

cairnswk wrote:The info for each two lines is also a little confusing.

Bonus Area Size 4
Terrs Required for Bonus 4

What does bonus area size actually mean?
Terrs Required for Bonus is self explanatory.

I think bonus area size means how much is big your bonus (number of total regions), territories required is how many of these territories into the area bonus are required to gain a bonus (i.e. <required> tag)
As you certainly know there're maps on which a continent A is composed by X regions on the whole, but you need just Y to receive a bonus (in general a partial bonus)
So the tool is done to take in account this.
In fact if you have a map with 42 regions and a bonus onto it that is a 4 regions bonus that works in this way "hold 4 for +5" and "hold at least 3 for +3" (just an example), then you have two very different percentages in the two cases.
When you receive 4 random regions on 42, you have the 1,79% of probabilities to have any player to start with that bonus if it's necessary to have those exact 4 regions.
Instead if you need just 3 of them to collect the bonus (don't care if the 4th is neutral or not), then you have a big 20% of probabilities of any player to start with it.
So set the area size (if there's one) is really important. It's not just 3/42, but it's ((3/4)/42) so ((required territories/area size)/total regions on the map)

Hope it helps,
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### Re: [Excel] Bonus Probabilities

^^ thanks tnb80

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