Five card draw card removal effects

[LLCoolDave]

Interesting concept you are analyzing there, havn't had the time to read it properly yet. What stuck out was this, though:

"since holding one pair slightly decreases your opponents' chance of having two pair, trips, and full house relative to if you held no pair (I think)"

Actually, I somehow think it's the other way around. If you hold a pair, there are 3 ranks he's less likely to have a pair and one where he is very unlikely to have a pair, where as holding 5 unpaired cards gives 5 ranks that are unlikely to pair up for an opponent. This is just an instinctive feeling, I havn't done the math to back this up, so I could very well be wrong. In any case, I think the difference is pretty minor.

Some of this math could very well help me play QQ and JJ better than I currently do, I still feel very unsure when playing these hands. I will take a closer look at this later.

[LearnedfromTV]

[ QUOTE ]
pair (u)......60198
pair (s)......32274
pair (i)......11559

u = unseen rank; s = seen rank; i = identical

[/ QUOTE ]

I see what I did wrong. I had these numbers at 84480/42240/14080. 84480 is of course the number of combinations of any pair for a randomly dealt hand (out of 52C5 possibilities). 6 ways to make a pair with 4 cards available, 3 if 3, 1 if 2, and I just divided the 84480 by the appropriate factor; but the other cards elminate some of the combinations also, i.e. JJ742, the number of combinations of 77xxx (in 47C5 given JJ742 dead)isn't just changed by the missing 7, but also the missing J, J, 4, 2.

I still don't understand why the ratios aren't 6:3:1. I guess if I have JJ742, an opponent with 77 and three random cards will have two pair relatively less often than one with JJ and three random cards? Because given that he has 77, JJ77 is tougher to make than if we give him JJ and try to make JJ77. Etc.

Thanks for the post, this is great stuff.

[cgull]

Bigpooch - Please write the book. I'll buy it. You consistently make wonderful posts on draw.

Since you know the game so well, do you still have fun playing ? Or is it a little mechanical ?

[AceOfClubs]

[ QUOTE ]
Bigpooch - Please write the book. I'll buy it. You consistently make wonderful posts on draw.


[/ QUOTE ]

[BBQbowser]

I am very big on this concept.

[mykey1961]

Here are some exact counts rather than estimates for what one opponent could have based on your 5 cards.

We have : JsJh4d3c2c

Opponent will have:
0.00001630 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183384 Flushes
0.00409338 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49887447 High Card
0.19500580 QQ+

0.29550436 One Pair+
0.00004153 One Pair=
0.70445411 One Pair-

0.79745870 We win
0.00001760 We tie
0.20252370 We lose


We have : AdJsJh3c2c

Opponent will have:
0.00001760 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183254 Flushes
0.00439457 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49857328 High Card
0.17710287 QQ+

0.23847251 One Pair+
0.00004153 One Pair=
0.76148597 One Pair-

0.82133057 We win
0.00001760 We tie
0.17865182 We lose

We have : AdKcJsJh2c

Opponent will have:
0.00001695 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183319 Flushes
0.00463513 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49833272 High Card
0.15913931 QQ+

0.19236714 One Pair+
0.00004153 One Pair=
0.80759133 One Pair-

0.84063186 We win
0.00001760 We tie
0.15935053 We lose


We have : AdKcQcJsJh

Opponent will have:
0.00001760 Straight Flush
0.00025229 Quads
0.00148050 Boats
0.00183254 Flushes
0.00473357 Straights
0.02156214 Trips
0.04803516 Two Pair
0.42385193 One Pair
0.49823428 High Card
0.14103364 QQ+

0.14891958 One Pair+
0.00004153 One Pair=
0.85103889 One Pair-

0.85894876 We win
0.00001760 We tie
0.14103364 We lose

[bigpooch]

Actually, these are estimates; exact numbers would be ratios
over C(47,5) = 1533939 and the relevant combinations are in
my previous post from which exact numbers can be obtained
when heads up.

These approximations seem correct except that it is
misleading for the ties to be exactly the same hand with the
same kickers because in practice, JJT42 and JJ943 have the
same pot equity when these hands are in (with the intention
of drawing three to a pair). Hence, the ratio of
11559/C(47,5) or about 0.007535501738 should be thought of
and not 27/C(47,5) or about 0.000017601743.

Also, once we have numbers for heads up, an approximation
should be used when considering more than two opponents
because the inclusion-exclusion calculation is horrendously
difficult for an exact probability that the hand is best
predraw; besides, because of the factor in the next
paragraph, it is sufficient to have a good approximation.

In addition, if there has already been some action, such as
one or two players have folded ahead of you, the ranks are
no longer equally distributed but rather very slightly
skewed towards kings and aces.