Bodiggy or whatever ...Badugi

-----------------------------------------------------------------------------------------------------------------------------------------

Anyway if you start with 2 good cards,... under your "target" say 7 high final hand... what are the probabilities, considering a RIGID "target" value, through all the redraws and stuff, of hitting your target hand......

well here are the results:

to the left clipped out of each image is the first draw... drawing 2 each time

red means you have a two card hand at this point

yellow means a 2 card hand

green means you have made a 4 card hand

13 is King high, 12 is Queen, 11 is Jack

if you don't know the game, I don't blame you, I still never played, and can't even spell it :-P

anyway the final numbers are a "success" summary ( making a 4 card hand, starting drawing 2, with a rigid target ( left column).... the second column is success % of that strategy, and the last column is the marginal

4	1.50%	1.50%
5	4.16%	2.66%
6	7.72%	3.56%
7	11.92%	4.20%
8	16.58%	4.66%
9	21.52%	4.94%
T	26.62%	5.10%
J	31.76%	5.14%
Q	36.86%	5.10%
K	41.85%	4.99%
		41.85%

the idea of the fixed target is unrealistic however, but the numbers are here and we can possibly make target decisions as we draw,... generally "widening" the target, to maximize success of completing a 4 card hand

-------------------------------------------------------------------------------------------------------------------

ok the question is now suppose we are dealt

should we hold the K or discard it?

let's look at what we are facing

if you hold the K, we will be drawing to a King high 4 card hand, drawing 1 on First Draw

the breakdown looks like this: simply you will make your hand 51.23% ( king high ) of the time

now if you discard the hand and Draw to a King high 4 card hand, ( now drawing 2 on the First Draw)... you will end up with a 4 card hand 41.85% ( king high or less ) of the time...

you will also end up with a 2 card hand 1.88% of the time ( worse than keeping the king by far )

now by sacrificing 51.23 - 41.85 = 9.38% in absolute terms, and 9.38%/51.23% or 18.3 percent of your successes of making a 4 card hand, but a likely better 4 card hand when you do

-------------------------------------------------------------------------------------------------------------------

here are the "Good Tables"

P| 2, *, * Probability Drawing for FIRST draw 2 cards... fixed target, ( no dead outs )

this means drawing possibly on all three draws, with the rigid target number

P| 2,2,* Probability Drawing 2 at Second Draw.... fixed target, no dead outs

P|2,1,* Probability Drawing 1 at Second Draw, after drawing 2 on First... fixed target, no dead outs

P|1,*,* Probability drawing 1 on First Draw

P| 1, 1, * Probability drawing 1 on Second Draw, after drawing 1 on First Draw

dead outs - are cards you discarded earlier, that are now in your enlarged target range

---------------------------------------------------------------------------------------------------

Question:

what happens if you draw one by discarding the King.

Are you any better off given that you then have the option to discard either the or if you happen to catch a ,,, .

Does this increase your chance of making an 8 low ?

Doubtful, but lets see what happens when we do the math

Original hand:

discard , and hold !!!!! Drawing 1 at First Draw

at the first draw there would be 4 outs on a 2 or 3, and 2*(t#-3) outs on a "solid hit"... here we use t#= 8 so 2*(t#-3) = 10... 4s 4c 5s 5c 6s 6c 7s 7c 8s 8c

having seen 4 cards, you will be drawing from 52-4 or 48 cards, so each out is 1/48 likely... there for there is a

4/48 chance you hit a 2 or 3

and a

10/48 chance we hit a "solid hit" ( although if you drew a 9, you might adjust your target to include it.. arguably)

now in the 4/48 chance you hit your 2 or 3... you will need to draw 1 to a 8 target card 8-3 = 5 so 5/47 you hit ( on the second draw ) and 5/46 on the third draw... this mean you will have a 5/47 + (42/47)*(5/46) = 20.35%chance of completing a 4 card hand 8 high or LESS from this position ( which only occurs 4/48 times at the second draw

now supposing you make a "solid hit".... 10/48 chance... we will then discard the 3 on the second draw.... 47 cards left.. and again there are only 5 ranks and one suit to complete.... so the math works out the same completing 20.35% of the time from Drawing one at an 8 high ,4 card hand at the Second Draw

now (48 - 10 - 4)/48 = 34/48 times we "miss" on the first draw not pairing a 2 or 3 and no solid hit 70.83%...

here we would need a decision to draw 1 or 2, and I am not sure which would make since, either abandon the "strategy" ( draw 2 ), or stick to your guns ( draw 1)... but certainly we are not in a good spot here,... holding Ah 2d 3d X at the second draw

if we drew 2 at the First Draw, we would have a 16.58% chance of hitting an 8 high 4 card hand

drawing 1 at First we have success = 4/48*.2035 + 10/48*.2035 + 34/48*X.... which would need to be > .1658 to be a dominant strategy ( with a fixed target 8 ) This yields X > 15.03%... so you would need a 15% chance or better to win after "missing" the First Draw

now if you abandon the strategy and begin to draw 2on the second draw, you will hit 9.49% of the time, 5.54% too low to make this a winning strategy

the only other option is to again fold the "missed" card..... 4/47 you hit your 2 or 3 (nd 5/46 on the third draw) and 10/47 on the "solid out", leaving you 5/46 chance of completing on Third

so 4/47*5/46 = 0.925%

and 10/47*5/46 = 2.31%

missing again on second, would make you draw two on the third draw... and the chance of hitting both would be just 30 in 1035 or 2.90%

so an additional 32/46*.029 = 2.02%

now we add .925 + 2.31 + 2.02 and get 5.26%... far from the 15.03% we needed,

working it backwards... having missed the first draw, we would draw 2 on the second draw, and before that, we should realize that drawing 2 on First gives us an over all better chance to hit

clear as mud right?

Clearly state, keeping two of one suit is not mathematically correct,... except that there might be some benefit through deceit of drawing 1 rather than 2, it sends a signal that you are ahead in the drawing round

-------------------------------------------------------------------------------------------

ok lets look at, betting in a very simple way 4 people to the First Draw all the way to the end,

you would need to pay 1/2, 1/2, 1, and 1 or 3 total, and the pot would have 12 after the last draw ( is this right ? )

anyway if you got any hand, you would want to win at least 25% of the time....

So Lets look at Ah 2d 3s Kh

assuming we draw 1 at the first Draw, we have a

K	51.23%
Q	47.16%
J	42.88%
T	38.37%
9	33.63%
8	28.65%
7	23.43%
6	17.96%
5	12.23%
4	6.25%

chance if hitting these Badugi's drawing 1 on First, so likely we would target 8 high, a likely best hand

if we would want to win 5 handed, we would want to choose a target of 7 high maybe, a stronger hand, in a bigger field

now betting is not this simple

suppose you can fold but your three opponents cannot... your chance of completing an 8 high goes from 28.65 to 20.35% Drawing still 1 on Second Draw and it happens 89.58% of the time, but facing a 1/2 bet, you are getting a chance at now 4, and soon 12... actually you will have to pay 2.5 to get the 12 or 20.8% of the pot hmmm, certainly these rules are too too simple LOL

-----------------------------------------------------------------------------------------------------------------

ok I just had a realization.... the way to think of this stuff is that... you are always drawing to the King,... but face a decision after you draw, at least in some instances

suppose you have:

looks amazing until you notice the suits, so looks like we keep , and draw 2 on the first betting round, we will just say conveniently, we were in the big blind and it checked to us,....

Now, we are looking at Drawing 2 in the First Draw, Drawing to a King, and actually we get a pdf... since we can get a 4 high Badugi at this point, we can catch a Badugi of king high through 4high,... and we have the numbers to show this pdf, probability distribution function... here we are dealing with discrete data, so it is basically a list, with numbers, that add to 1, or 100%.... ( if you include all possibilities, such as missing)

Now at each point in this pdf, we face a decision, that will have an answer based on the pdf, that results from it...

To reduce the nesting some, here we will just say we draw, at the First Draw, , ....

giving us

clearly we need to discard the since it matches suit with a lower rank ( Although there may be a real decision here as far as dead cards ??... suppose you already saw 1, earlier... I have to examine this later ... )

Anyway, do we keep the king??? And we will get a not so clear, but very well defined answer.... two separate pdfs that we can compare.... if we keep the king, we will be drawing 1 at the Second draw, trying to hit a king high Bodiggy thinggy... now here is the "trick", in this instance... the pdf has all it's mass for success on one spot.. king high... since we have a king in our hand, we can get no less than a king high...

looking at my charts we have a 32.42% of catching a king high Badugi drawing 1 at Second... this includes the chance to hit on Third.... so here is the pdf of the result of this decision:

King high: 32.42%

missed : 67.58%

Now our other option is: Drawing 2 on Second for a King high Dagpuppy, ... which yields.. pretty sure,...

K	4.04%
Q	3.93%
J	3.75%
T	3.53%
9	3.24%
8	2.89%
7	2.47%
6	1.98%
5	1.40%
4	0.75%
	27.98%

so we sacrifice the better chance of success 32%, for a lower 28%, but we have a 6.6% chance now of catching a 7 high ( or lower) Badugi... we have a 16.26% chance of getting a Ten high DaBugi, and we have a 4.04% chance of hitting that same King high ( Actually, I need to add in the "dead card thing, since one of our outs is burned here... so this pdf is slightly off)

now... it is important to say, that suck lists like this in this form, discrete pdf and so forth, have some really good features... like we could determine the "middle"... which is half way down ( 27.98% / 2 ) or 13.99% counting from kings down, or 4's up... which puts us solid in the Ten high row... meaning ... this is the mean... notice the mode ( row that has the biggest number ), is king high, such that we have some skew here

Now... should we have some heuristic to assign values to these probabilities, we will get solid yes no decisions.... we should determine a crappy one first,... and then we can make it better.... as you said, if we have hand EV's, for after showdown hands, drawn as data, the kind I proposed for your all in strategy, centered on 0, we can determine the best decision as far as drawing is concerned

--------------------------------------------------------------------------------

The odds of getting dealt a Badugi on your first 4 cards

(13 choose 1)(12 choose 1)( 11 choose 1)(10 choose 1) /( 52 choose 4) = 6.34%

this is the chance of a king high or lower badugi

by changing the 13 to 12 and so forth, we can determine the chance of hitting lower

generalizing .....

(T choose 1)(T-1 choose 1)( T-2 choose 1)(T-3 choose 1) /( 52 choose 4) for 4<= T <= 13

this can be rewritten:

T!/(T-4)! / (52 choose 4)

high card	or lower	exactly
K	6.34%	1.95%
Q	4.39%	1.46%
J	2.93%	1.06%
T	1.86%	0.74%
9	1.12%	0.50%
8	0.62%	0.31%
7	0.31%	0.18%
6	0.13%	0.09%
5	0.04%	0.04%
4	0.01%	0.01%
		6.34%

the exactly column, is marginal probability that you hit exactly a king high badugi (and not a 7 high one for example)... we are removing the or lower part basically
notice that half of them are in fact Queen high or higher
notice 8 high hand ( or lower ) is less than 1%, and only 9.8% of pat badugi's, so 90% of the time they will have no lower than a 9 high

-----------------------------------------------------------------------------------
sample data

	4 card	3 card	2 card	1 card
K	1	3	1
Q	1	3
J		4	1
T	1	5	2
9	1	5	5
8		2	1
7		1	2	1
6		2	3
5		1
4			2
3			2
2
1
T	4	26	19	1

so 4 card Badugi is 8% centered on J
a 3 card hand is 52% centered on T
2 card is 38% centered on 9
and a 1 card it was all the same suit not rank, 2% centered on 7?

this is real deck simulation, with 50 hand dealt hands, no Draws, just 4 cards

the sample size very small, but it should give us a good idea about the distributions