Help Wanted With Mike Caro's Misery Index (Was Re: Raising Pair on the Button)
Alan Bostick
"wdcowey" <wdc...@diversicomm.com> wrote:
> Mike Caro published a chart he called the "misery index". For any given
> pocket pair it is the probability that you will flop at least one overcard
> and not have a set or quads. Here is the chart: I have included a 2nd
> column that gives the probability of at least one overcard, and does not
> take sets and quads into account.
>
> Pair Percentage Percentage (2)
> AA 0 0
> KK 11.67 22.5
> QQ 31.12 41.43
> JJ 46.96 56.96
> TT 59.67 69.49
> 99 69.30 79.29
> 88 76.54 86.74
> 77 81.65 92.14
> 66 85.01 95.84
> 55 86.97 98.14
> 44 87.92 99.39
> 33 88.22 99.90
> 22 88.24 100
I need some help with a Yooner. I've been figuring something that I
think ought to be the same thing, but my numbers don't agree. Would
anyone (Barbara? Mike Caro?) care to critique my reasoning?
Suppose you hold some pocket pair. The probability that you'll flop a
set (including those times you'll flop a full house) is going to be
P(set) = 3 * 2/50 * 48/49 * 47/48 = 282/2450 (= 13536/117600),
and the probability that you'll flop quads is
P(quads) = 3 * 2/50 * 1/49 * 48/48 = 6/2450 ( = 288/117600).
Now, call the number of ranks less than the rank of your pair R. (For
aces, R=12; for deuces, R=0.)
The probability that your pocket pair is an overpair to the flop is
the probability that all three cards are of rank less than your pair.
There are four cards for each such rank, and the overall probability is:
P(overpair) = 4*R/50 * (4*R - 1)/49 * (4*R - 2)/48
= 4*R*(4*R - 1)*(4*R - 2)/117600.
And so the probability that the flop will be "good" for your pair is the
sum of P(overpair) + P(set) + P(quads).[1]
P("good") = (4*R*(4*R - 1)*(4*R - 2) + 13536 + 288)/117600
And so the remaining flops are "bad" flops, ones in which you don't flop
a set or quads and which do contain overcards: P("bad") = 1 - P("good").
Here is a tabulation of P("bad") for pocket pairs:
Pair R P("bad")
AA 12 0.0000
KK 11 0.2067
QQ 10 0.3784
JJ 9 0.5182
TT 8 0.6294
99 7 0.7153
88 6 0.7792
77 5 0.8243
66 4 0.8539
55 3 0.8712
44 2 0.8796
33 1 0.8822
22 0 0.8824
Now I have tried to construct P("bad") to be what Wayne Cowey describes
as Mike Caro's Misery Index: "the probability that you will flop at
least one overcard and not have a set or quads." And yet my numbers
don't agree with the numbers Wayne quotes as being Caro's; mine are
higher except for 22 and 33; and the difference grows as card rank
goes up.
Here are some guesses for the explanation of the discrepancy:
(1) I've done my math wrong. (It wouldn't be the first time.)
(2) Mike Caro did his math wrong. (Hey, even Homer[2] nods.)
(3) Mike's Misery Index accounts for things that my calculation doesn't
-- a flop containing an underpair is dangerous, and those are certainly
countable.
Comments and criticism greatly appreciated!
My motivation for figuring this out was to get some insight on the
problem of playing pocket pairs. You want to raise with big pairs and
limp in with small pairs; I wanted and want some sense of where the
break point between the two regimes is. Then, after figuring this out,
I remembered about the Misery Index and looked up Wayne's post.
[1] Many of these "good" flops are in fact trouble flops -- flops
containing a pair, three to a straight, three of the same suit, etc. If
you hold Ah Ad and the flop comes Js Jc Ts your hand is in serious trouble.
[2] The Greek poet, not the American Simpson.
--
| Many that live deserve death. And some that die
Alan Bostick | deserve life. Can you give it to them? Then do
mailto:abos...@netcom.com | not be too eager to deal out death in judgment.
news:alt.grelb | J. R. R. Tolkien
http://www.alumni.caltech.edu/~abostick
Barbara Yoon
> ...I have tried to construct...what Wayne Cowey describes as Mike Caro's
> Misery Index [for starting pairs]: "the probability that you will flop at least
> one overcard and not have a set or quads." And yet my numbers don't
> agree with the numbers Wayne quotes as being Caro's; mine are higher
> except for 22 and 33; and the difference grows as card rank goes up.
Likely a difference in how Caro (via Cowey) defines/calculates it...and a clue
(though haven't followed through on it) in how your numbers are the same for
2-2 and 3-3, both of you treating starting 3-3 and a flop of 2-2-2 the same as
NON-misery...
> Mike's Misery Index accounts for things that my calculation doesn't...
...or vice-versa...
Joesmallie
>From: abos...@netcom.com (Alan Bostick)
<<And so the probability that the flop will be "good" for your pair is the
sum of P(overpair) + P(set) + P(quads).[1]>>
What about when you flop an open ended straight with a pocket pair?
Mike Caro
You may be thinking that the hold 'em "Misery Index" is more
sophisticated than it really is. It was presented purely to drive home
a point. It is not intended to be an accurate gauge how much trouble
you're in.
The figures meet the definition of "misery" provided, but you need to
"suspend your sense of disbelief" in order to accept that simple
definition as appropriate for all situations. For instance, in some
situations you may be relatively happy with K-K and a flop such as,
A-9-4, which is defined as a miserable outcome. And you may be
relatively apprehensive with 3-3 and the flop 2-2-2, even though this
does not meet the "miserable" definition.
The table thinks that when you hold a pocket pair, every flop is
miserable unless (1) You catch at least one additional rank
complementing the pair, or (2) all three flopped cards are lower in
rank than your pair. It ignores the fact that with Q-Q, a flop like
A-A-Q leaves you in jeopardy, and simply thinks that's non-miserable.
With a pair of deuces, you can't have three cards lower than your
pair, so you will meet the misery index definition 88.24 percent of
the time -- which corresponds exactly to your chances of catching at
least one deuce on the flop.
With a pair of threes, you add about 1/50th of one percent to the
probability of seeing at least one more three on the flop. In other
words, IF you don't flop a three, then what are the chances that all
the ranks will be lower than your pair? In the case of a pair of
threes, ONLY 2-2-2 could meet that definition. The chances of that are
precisely one in 4,900 -- or about 0.02 percent -- which is why the
misery index for 3-3, at 88.22 percent, is not as great as the 88.24
percent for 2-2.
Due to the volume of statistics that I've published, and my aversion
to triple checking, it wouldn't surprise me (well, maybe a little :-))
if some numbers, somewhere are incorrect. My track record has been
extremely good (maybe I'm lucky), but if you want to put even more
confidence in published figures, try W. Lawrence Hill. He rigorously
checks and rechecks everything. Me, I like to gamble. So, I appreciate
the scrutiny.
Straight Flushes,
Mike Caro
On Sun, 12 Jul 1998 22:57:15 GMT, abos...@netcom.com (Alan Bostick)
wrote:
>And so the probability that the flop will be "good" for your pair is the
>sum of P(overpair) + P(set) + P(quads).[1]
>
>P("good") = (4*R*(4*R - 1)*(4*R - 2) + 13536 + 288)/117600
>
>And so the remaining flops are "bad" flops, ones in which you don't flop
>a set or quads and which do contain overcards: P("bad") = 1 - P("good").
>
>Here is a tabulation of P("bad") for pocket pairs:
>
>Pair R P("bad")
>
>AA 12 0.0000
>KK 11 0.2067
>QQ 10 0.3784
>JJ 9 0.5182
>TT 8 0.6294
>99 7 0.7153
>88 6 0.7792
>77 5 0.8243
>66 4 0.8539
>55 3 0.8712
>44 2 0.8796
>33 1 0.8822
>22 0 0.8824
>
>Now I have tried to construct P("bad") to be what Wayne Cowey describes
>as Mike Caro's Misery Index: "the probability that you will flop at
>least one overcard and not have a set or quads." And yet my numbers
>don't agree with the numbers Wayne quotes as being Caro's; mine are
>higher except for 22 and 33; and the difference grows as card rank
>goes up.
>
Alan Bostick
joesm...@aol.com (Joesmallie) wrote:
>
> >From: abos...@netcom.com (Alan Bostick)
>
> <<And so the probability that the flop will be "good" for your pair is the
> sum of P(overpair) + P(set) + P(quads).[1]>>
>
That's why I put "good" and "bad" in quotes. Some of the "good" flops
are actually quite bad (look at Mike's AAQ when holding QQ). Naturally,
some of the "bad" flops are pretty damn good (e.g. Ah Kh Qh when
holding Jh Js).
Winner777
with a flop of AhKhQh.
Ed Hill
Mike Caro
Just to clarify, I didn't say that A-A-Q was a bad flop for Q-Q. It is
much better than average outcome in terms of potential profit,
especially in limit games. I said that the method behind the "Misery
Index" ignored the jeopardy that exists. The index treats A-A-Q and
Q-J-J as equally non-miserable flops for a pair of queens. That was my
point. This was probably not the best example, though.
Straight Flushes,
Mike Caro