Favorite/Underdog

[Apocalypse]

In "Winning Low-Limit Hold-em" by Lee Jones there is an introductary passage of playing the flop. He gives the following example:

You hold: As Ks

Your opponent holds: 8c 7d

On these flops there are various odds:

Before the flop : you are 2:1 favorite
Flop is Qd 9h 2c : you are 3:1 favorite
Flop is Qd 8h 2c : he is a 3.2 :1 favorite
Flop is Qd 8s 2c : he is a 2.5 :1 favorite
Flop is Qs 8s 2c : you are even money (1:1)
Flop is Ad 6h 2h: you are a 15:1 favorite
Flop is Ad 8h 4c: you are a 4:1 favorite
Flop is Qd 8h 7c: you are a 19:1 favorite
Flop is Ad Kc 2h: you are a 164:1 favorite

My question is actually really simple: what are the calculations behind this?? I tried some stuff, but it never gave the results presented here. If someone would have the kindness of showing how its done (especially the even money thing), id greatly appreciate it. Thanks! /images/graemlins/smile.gif

[bigpooch]

You hold: As Ks
Your opponent holds: 8c 7d
On these flops there are various odds:

Before the flop : you are 2:1 favorite

The calculations for before the flop are very tedious and I
won't go into details but defer to the twodimes analyzer:

http://twodimes.net/h/?z=13305
pokenum -h as ks - 8c 7d
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
As Ks 1100896 64.29 605049 35.34 6359 0.37 0.645
8c 7d 605049 35.34 1100896 64.29 6359 0.37 0.355


For the calculations after the flop, the key number to think
of is C(45,2) = 990 or the possible combinations of turn and
river cards (this is just 45x44/2 as the order of the turn
and river cards won't matter as to who wins the pot). Thus,
it's simply the matter of enumerating the combinations that
win for one of the hands. Then, you only need to look at
wins:losses = wins:(990-wins) to get a ratio for the odds.
Also, the hand to look at that is often easier to make the
calculation for is the worse hand on the flop. As a double
check, you can go to twodimes.net but that would not be as
instructive.

Flop is Qd 9h 2c : you are 3:1 favorite
The 87 can make trips or two pairs in 3+3+3x3=15 ways.
The 87 can also make a pair in 6x(36-3)=198 ways; here 36 is
just the cards other than an A, K, 8 or 7; since 3 of these
cards are on the flop, these cards cannot appear on the turn
and river and are subtracted. Finally, the 87 can make two
possible straights, each yielding 4x4=16 combinations and
altogether there are 15+198+32=245 of them and hence the 87
is a 3 2/49 to 1 or 3.040816 to 1 dog.

Flop is Qd 8h 2c : he is a 3.2 :1 favorite
The AK can make trips or two pairs in 15 ways.
Now, the AK can also make a pair in 6x(36-2)=204 ways.
There are also 16 combinations of a big straight so in all
there are 15+204+16=235 combinations and so the AK is a
3 10/47 to 1 or 3.212765 to 1 dog.

Flop is Qd 8s 2c : he is a 2.5 :1 favorite
This is just almost the same as the above except that the
AK can now make a back door flush. There are C(10,2)=45 of
these combinations including precisely Js Ts which was
included in one of the wins in the previous example; i.e.,
the number of wins for the AK now increases by 45-1=44 and
so there are now 279 combinations yielding odds of 2 17/31
to 1 or 2.548387 to 1.

Flop is Qs 8s 2c : you are even money (1:1)
The As Ks could make a flush: two spades coming down give
C(9,2)=36 ways and just one spade results in 9(39-3)=324
ways but not all of these win! The 7s plus another 8 or 7
occurs in 4 ways and the 2s and another 8 or 2 occurs also
in 4 ways so a total of 8 combinations result in the 87
winning so the number of flush wins = 36+324-8=352. There
are also 3x3=9 straights and 15 trips or two pairs. Lastly,
hitting an A/K and then a nonflush card with a rank that is
different from A, K, 8 or 7 gives 6x(3x9-1)=156 combinations
(the 2c is on board so this one card can't be seen later).
Thus, there are a total of 352+9+15+156=532 combinations and
so the AK is a 1 37/229 to 1 or about a 1.161572 to 1
favorite.

Flop is Ad 6h 2h: you are a 15:1 favorite
This one is easy: 87 can make any of 3 straights (16 ways
each) and trips or two pairs (15 ways) to win or in 48+15=63
ways and thus, is a 14 5/7 to 1 or 14.71429 to 1 dog.

Flop is Ad 8h 4c: you are a 4:1 favorite
The 87 can make quads in 1 way, a full house in 3+2x3=9
ways, a straight (by a 6 and a 5) in 16 ways. For trips,
there are 2x(40-2)=76 wins (not an A/8/7 and no Ks, 4c) and
for two pairs there are 3x32=96 wins (8 ranks of other cards
allowed: no A/K/8/7/4 for the 87 to win and not included in
one of the above wins). Thus, the total is 1+9+16+76+96=198
indicating that the AK is exactly a 4 to 1 favorite.

Flop is Qd 8h 7c: you are a 19:1 favorite
You must have entered this incorrectly. It should say
HE is a 19:1 favorite. The calculation is very easy: AK can
only win by making trips or ace and kings with 15 ways or
aces and queens or kings and queens in another 6x3=18 ways.
A straight can also be made in 16 ways so the total wins are
15+18+16=49 so that the AK is a 19 10/49 to 1 or 19.20408 to
1 dog.

Flop is Ad Kc 2h: you are a 164:1 favorite
The 87 can only win by making trips and that's done in just
3+3=6 ways and thus, the 87 is a 164 to 1 dog.

These calculations won't necessarily help your play though!
/images/graemlins/smile.gif

[Apocalypse]

i see said the blind man....

WOW and THANKS!!

and no it won't help me play better, but working and getting sense with the numbers does sharpen and develop my view thus decisionmaking in a hand. It creates 'feel' whatever that exactly mean... thanks! /images/graemlins/smile.gif

[bigpooch]

Actually, the 87 can also make bottom straight when a Jack
and Ten hit the table, but of course, the AK can then make
nut straight. So, I need to say that the 87 can make two
types of winning straights and 1 losing straight! The
calculation is nevertheless correct.