odds on a player having a set

by **Bonedancer** on Wed Nov 21, 2007 10:31 am

Are there any mathmatics whizzes out there that can answer a few questions about cards for me?

if a player has three cards, what are the odds that he has a set to turn in?
if a player has four cards, what are the odds that he has a set to turn in?

Thanks a bunch,

Bonedancer

by **Velvecarrots** on Wed Nov 21, 2007 10:46 am

3 cards - 1/3 chance of having a set. No matter what a player's first 2 cards are, he has to get a certain color (1/3) to make a match.

As for 4 cards, there is a 7/9 chance that you have a set. (Although if you don't have a set after 3 cards, there is a 2/3 chance you get a set with your 4th card). If you don't have a set with 3 cards, then no matter what the colors of your cards are, 2 out of the 3 colors will make a set with 4 cards.

by **MeDeFe** on Wed Nov 21, 2007 1:24 pm

As for mixed sets, for three cards the chance is 2/9, 2/3 for four cards and 7/9 for five cards. Unless I've made a mistake somewhere, but I don't think so.

by **Piestar** on Wed Nov 21, 2007 5:28 pm

Re: 5 card sets, if you discount all hands where you had a set with 3, or had a set with 4, you have a 1 in three chance of pulling a mixed set with your fifth card.

To have 4 and no mixed set, you have to have two and two. (the color matters not) You fifth card has a 1/3 chance of being your first color, a 1/3 chance of being your second color, and a 1/3 chance of being the third color, that needed to make a mixed match.

Curious, was there ever any serious discussion regarding my idea for alternet decks?

My Conquer Club Scripts · by **Aerial Attack** on Wed Nov 21, 2007 6:05 pm

The chances for having a set are as follows:

Red, Green, OR Blue: 3 Cards (1/27 = 03.7%), 4 Cards (9/81 = 11.1%), 5 Cards (31/243 = 12.8%)

Mixed Colors: 3 Cards (6/27 = 22.2%), 4 Cards (36/81 = 44.4%), 5 Cards (150/243 = 61.7%)

Escalating (Specific Colors don't matter): 3 Cards (9/27 = 33.3%), 4 Cards (63/81 = 77.7%), 5 Cards (243/243 = 100%)

EDIT: Made all percentages to the tenth of a percent (for easier summing). Added to the following two percentages:

4 Cards All the Same Color: 3/81 (03.7%), so you know now that you are NOT going to get a mixed set.

Chances of pulling "runner runner" cards to turn a red set into a mixed set = 2/9 (22.2%)

by **Piestar** on Wed Nov 21, 2007 11:29 pm

Aerial Attack wrote:The chances for having a set are as follows:

Red, Green, OR Blue: 3 Cards (1/27 = 04%), 4 Cards (9/81 = 11%), 5 Cards (31/243 = 13%)

Mixed Colors: 3 Cards (6/27 = 22%), 4 Cards (36/81 = 44%), 5 Cards (150/243 = 62%)

Escalating (Specific Colors don't matter): 3 Cards (9/27 = 33%), 4 Cards (63/81 = 78%), 5 Cards (243/243 = 100%)

I have to suggest your math is flawed AA. 4% of a matched set with three, but 22% for a mixed set with three? Think about it.

Forget the colors, with three cards, there are only nine real outcomes.

Whatever your first card color is, this your base color. Red, Blue, Green, it doesn't affect your odds.

Your second card allows for three possible outcomes. One where the colors are the same, and two with the colors not-the-same.

Each of these three also has three potential outcomes.

Your nine potential outcomes are as follows

111
112
113
121
122
123
131
132
133

Total of nine potential variations., regardless of what color one and color two are. (Or color three of course)

There is no need to start a set starting with color 2, because color one is defined as 'the color you get on your first card.'

One in nine (111)can be a three color match (about 11%), two in nine (123 and 132) can be a mixed color match, or about 22%.

Four cards has a potential of 27 outcomes, and five cards has the potential of 81 outcomes, but for real world purposes a lot of those outcomes are irrelevant.

In Escalating, you want to wait as long as possible to turn in your first set, to maximize it's value, after that you generally want to turn in it as soon as possible to avoid becoming a valuable target for having a lot of cards.

In flat rate however, players might want to know if it's worth waiting for that extra card, and getting a more lucrative turn in.

If you have a set of three the same color, obviously the fourth card has a one in three chance of being the same color, if this happens you should go ahead and turn in. This would give you a zero percent chance of a mixed set at five cards.

If your fourth card is a different color (2/3rds chance) then you have a one-in-three chance of getting the missing color, and a two-in-three chance of matching one of the two colors you already hold.

It seems to me that waiting for the purpose of getting a mixed set is not a good statistical option (success one time in three), unless you have no use for the armies the cards would produce right now. Again, you don't want to be a target late in the game, because you have a lot of cards, so there is motive to turn in, if you are near elimination.

Conversly, it is almost worthwhile to intentionally have no cards near the end of the game, to make you less 'tasty'. Consider this if your position is poor, yet you feel the need to attack out of habit. It's not always the best course.

by **MeDeFe** on Thu Nov 22, 2007 7:15 am

He meant 4% for each colour, so 4% for a red, 4% for a green and 4% for a blue set.

by **Piestar** on Thu Nov 22, 2007 2:48 pm

MeDeFe wrote:He meant 4% for each colour, so 4% for a red, 4% for a green and 4% for a blue set.

Really? How bizarre... maybe so, the numbers do add up. Seems like he did three times the math required to get there then. Did he think the odds for three reds might differ from three greens, or three blues?

by **wacicha** on Thu Nov 22, 2007 3:33 pm

Unless you are playing Wacicha, if he has 3 cards he has a set 95% of time I love the card gods right up there with the dice gods...

by **Piestar** on Fri Nov 23, 2007 12:34 am

Good to know, better put you on my ignore list... tough enough to win as it is, without playing a person Blessed by the Gods of Dice, AND the Gods of Cards...

:C)

by **Richard Hand** on Thu Aug 07, 2008 7:01 am

Anyone know what the seemingly random numbers BOB throws out there under the column "card set estimate" ?

by **Kemmler** on Thu Aug 07, 2008 7:03 am

Richard Hand wrote:Anyone know what the seemingly random numbers BOB throws out there under the column "card set estimate" ?

it's some complicated formula that no-one understands, I've done a forum search with not much luck.

by **banana_hammocks** on Thu Aug 07, 2008 3:37 pm

Kemmler wrote:
Richard Hand wrote:Anyone know what the seemingly random numbers BOB throws out there under the column "card set estimate" ?

it's some complicated formula that no-one understands, I've done a forum search with not much luck.

I have always assumed it is based on "the likelyhood of them having each set (as shown above) * value of the set (4 red, 6 green) + % of board they own * 6."

the reason for the % of board they own * 6 is because you get 2 auto deployed if you own the territory on the card and 3 cards are being turned in.

I haven't checked this but it seems to me likely (ie. if 2 players have the same number of cards the one with the more territories has a slightly higher number.)

I will go check if this is correct with some basic guesses.

EDIT:
from Game 3001666 where 1 player has 3 cards (escalating value 4) and 16/36 of the territories. Bob gives a value of 2.22.

which works out as :
chance of having a set (9/27) * value of set (4) + %of territories owned (16/36) * 2 = 2.22222222222

So the end is * 2 for the bonus additions although i'm not sure that is right as if you own all the map and are trading in 3 cards you will get 2 bonus troops on each of the 3 territories not just 1. Equally if you own half the map you would expect to get 3 troops not 1.

EDIT 2:

with 5 cards it is * 6 rather than * 2