a little 2-7 TD math

[NYplayer]

I want to know the chances of someone improving to 9-7 or better under the following conditions.
-They start with 2-3 and draw 3.

-on after the first draw they will fold if they dont improve to a 2 card draw 8 high or better.

- after the second draw they will fold if they don't improve to a 1 card 9 draw or better.

i started to do this math but percentages were not adding up to 100%. for example, on just the first draw alone i ended up eith him not improving at all 18% of the time, impriving to a 2 card draw 54% of the time, imroving to a 1 card darw 27% of the time and improving to an 8 4% of the time. this adds up to more than 100% though [img]/images/graemlins/confused.gif[/img]

can someone try this calculation for the first draw and let me know how often they get each of those events happening?

[MarkGritter]

Your question is a bit ambiguous.

What is the set of second round hands you have in mind? It includes:
Pat 7
Pat 8
1-card draw to a 7
1-card draw to an 8
2-card draw to a 7
2-card draw to an 8

What about:
1-card draw to a 9?
pat 9?
(some of which may be worse than the 97xxx target, but people have been known to make such mistakes.)

[MarkGritter]

Also, do you want an answer with or without any additional cards (like a pat 97654 they are drawing to beat, for example?)

[NYplayer]

[ QUOTE ]
Your question is a bit ambiguous.

What is the set of second round hands you have in mind? It includes:
Pat 7
Pat 8
1-card draw to a 7
1-card draw to an 8
2-card draw to a 7
2-card draw to an 8

What about:
1-card draw to a 9?
pat 9?
(some of which may be worse than the 97xxx target, but people have been known to make such mistakes.)

[/ QUOTE ]

you are correct, i wasn't as specific as i should have been. if he gets a pat 9-7 or better he stands, otherwise he draws. he will keep a 1 card draw to a 9-7 or better.

[NYplayer]

[ QUOTE ]
Also, do you want an answer with or without any additional cards (like a pat 97654 they are drawing to beat, for example?)

[/ QUOTE ]

well, since the type of hand they are trying to beat can vary i think that having a specific hand will only muddle the math so assume it's against a random hand.

[MarkGritter]

OK... I started work before seeing your clarification, but fixing the numbers should not be hard.

Here's what I got by assuming that the initial hand was 27AKQ, we had no information about any other dead cards, and that we treated anything 9 and higher as a brick. (So, I left out pat 9s and 1-card draws to a 9.)

Summary:
Pat 7's: 256 out of 16215 draws, 1.6%
Pat 8's: 384 out of 16215 draws, 2.4%
1-card 7 draws: 2880 etc., 17.8%
1-card 8 draws: 1920, 11.8%
2-card draw: 7850, 48.4%
3 bricks (check): 2925, 18.0%.

This totals to the required 16215 = 47 choose 3 draws, so we know we've covered all the cases (unless I made two errors which cancelled out.)

More detail:

Calculation is by combinations. 7 draws = 543, 643, 653, 654. Each has 4 cards left for each rank = 64 combinations each = 256 of the draws.

8 draws = 843, 853, 854, 863, 864, 865. Same math.

1 card draws to a 7 = two babies and one brick. The example combination 43x. X can be a 9 or higher, a 2, a 3, a 4, or a 7. These cases give 336, 48, 24, 24, and 48 combinations respectively, totalling 480 combinations. Repeat for 53x, 54x, 63x, 64x, 65x = 2880 combinations total.

1 cards draws to an 8: works out just like the previous one--- consider 83x, 84x, 85x, 86x. 480 * 4 = 1920.

2 card draws. baby-brick-brick. The example is 3xy. Let B=9+, then the combinations are 3BB, 3B7, 3B3, 3B2, 377, 373, 372, 333, 332, 322. Subtotal for each combination is 840, 252, 126, 252, 12, 18, 36, 4, 18, 12. (There are 21 cards 9 or greater left in the deck, so 3BB = 4 * 21C2, for example.) This gives 1570 combinations for 3xy, repeat for 4xy, 5xy, 6xy, and 8xy.

Not improving: brick-brick-brick may be BBB, BB2, BB7, B77, B72, B22, 777, 772, 722, 222.

[MarkGritter]

OK, and here we go keeping pat 97s and 1-card draws to 97s.

Again there are 16215 total 3-card combinations we could draw to 72AKQ.

Pat 7: 256, 1.6%
Pat 87: 384, 2.4%
Pat 97: 384, 2.4%
1-card draws to a 7: 2496, 15.4%
1-card draws to an 87: 1920, 11.8%
1-card draws to a 97: 1664, 10.3%
2-card draws to a 7 or 8: 6186, 38.1%
3 bricks (check): 2925, 18.0%, total 100%

Details:

Pat 7s, 8s are the same. Pat 97 comes in six flavors: 943, 953, 954, 963, 964, 965.

1-card 7 draws are still 43x, 53x, etc. but now x must be a T or higher (otherwise we'd have a pat 9.) There are 17 T+ left in the deck, so there are, for example, 272 combinations 43,T+ that gives us a 1-card draw to a 7.

However, for 1-card eight draws, the "brick" is still 9 or higher or a pair, because we will break 98xxx.

The 1-card 97 draws are 93x, 94x, 95x, 96x, and x is T+, 2, 3, 7, or 9. These bricks give number of combinations 272, 48, 24, 48, and 24 respectively, so 416 cases for each resulting 1-card 97x2 draw. Thus 1664 cases total.

The 2-card draws are a little more complex than the previous example. For each of 3xy, 4xy, 5xy, and 6xy, the bricks may be T+. For 8xy the bricks are 9+ (again because we break a 98.) So the 3xy through 6xy total 1154 cases each, while 8xy is 1570 combinations.

Three bricks still come in the following flavors: 9+9+9+, 9+9+2, 9+9+7, 9+77, 9+72, 9+22, 777, 772, 722, 222. Note that we don't have to use T+ because none of these cases would give us a one-card draw to a 9, only a two-card draw.

[bigpooch]

Well, you might be aware of the limitations of the findings
of such a calculation. In any case, let me give a few brief
caveats for any attentive readers out there.

1) There is a bunching effect and it's not insignificant in
any lowball type of game. For example, there's a world of
difference when everyone takes the draw and when just the
small blind completes the bet in a five-handed game when you
are in the big blind with 32XYZ. When nobody's competing,
you are pretty sure the remainder of the deck is more likely
to be rich in baby cards; if everybody is taking the draw,
you may not have too many cards that make your hand.

2) We may as well assume the 3 and the 2 aren't suited and
you start with 32XYZ where the X, Y and Z are BRICKS (which
we may as well define as any ten to ace and they could also
be a deuce or a trey). Even if they are suited, clearly,
you will not end up with a flush MORE than 63/64 of the
time (it's more, because even if you did start off suited,
you could catch another deuce or three that makes you at the
very least unsuited).

3) This is not necessarily going to help you directly in
playing 2-7, but at least might give you some intuition or
feel about what to expect when you draw three cards to
something smooth. Obviously, if you start off with 33222
and see the other deuces and threes, you have an easy snow.

CALCULATIONS:

4) The calculations: There are C(47,3) = 16215 combinations
of draws. Note there are 23 "bricks" since you already
started with three of them and there are 20 aces down to
tens plus the six other deuces and treys.

a) Make a nine or better

Straight 6: 1*64 = 64
Pat 8 or better: 9*64 = 576
Pat 9 (better than 98): 4*C(4,2)*16 = 384
Pat 98: 4*64 = 256
(A pat 98 is just a special kind of draw to an 8 since the
objective is to make a 97 or better, just like the straight
six is a special kind of draw to an 8 or better.).

You make a pat 97 or better on 960 combinations or a
probability of about 0.059204.

b) Make a one-card draw to a nine or better

Draw to an 8 or better:

You can catch two of the five ranks plus a brick in
C(5,2)*16*23 = 3680 combinations. You can also pair up on
one of the ranks you catch in another C(5,2)*2*4*6 =
480 combinations for a total of 4160 combinations.

Draw to a 97/96/95/94:

You can see immediately that instead of the C(5,2)=10 above,
there are 4 types of hands, so it's just 4/10* the number
of draws to an 8 or better or 1664 combinations.

Draw to a 98:

Clearly, there are 416 of these.
(But these are essentially just a two card draw kind of
hand since the objective is to make a 97)

In any case, you will make a draw to a 97 or better in 64
(from the straight sixes) + 256 (pat 98s) + 4160 + 1664 =
6144 combinations or with a probability of about 0.378908.

c) Make a two-card draw to a nine or better

You can catch a four to a nine and two bricks in 6*4*C(23,2)
= 6072 ways or a pair of good cards and a brick in 6*6*23 =
828 ways or trips of a good card in 6*4 = 24 ways. In all,
there are 6924 of these combinations.

In any case, including the draws to a 98 in b), you get a
two card draw in 416+6924 = 7340 combinations and there is
a probability of about 0.452667.

d) You catch bricks

There are C(23,3) = 1771 combinations of these and the
chances are about 0.109220.


In any case, you can see that the total combinations add up
to 16215.


More?
-----

Is there much interest in TDL draw odds and an analysis of
the game? It seems fruitful to take a Monte Carlo approach
if you have some sort of poker dealing engine.

[MarkGritter]

Hm, I'm an idiot, I did all that for 27xxx instead of 23xxx. Oops.

[NYplayer]

[ QUOTE ]
Hm, I'm an idiot, I did all that for 27xxx instead of 23xxx. Oops.

[/ QUOTE ]

same difference. thanks for the math!