t4bz
> I don't see why holding a Jack and King of different suits would be
> better than holding a Jack, Ace, and King of different suits?
nudge51 replied:
You have to look at both of the P's. Possible and probable.
Naturally, with high cards the first thing you are looking for
is to pair, and push on the hand. Now if you look at possibles,
holding all three high cards, there is a zero possibility of drawing
three runners and hitting a quad, which is possible holding only two.
Also zero possibility of filling a full house. Holding two you can
draw a boat. Holding the three high cards, you can fill only one
straight, the ace high. Holding the J and K, you can fill two
straights, K high and A high.
You have nailed the essence, nudge, but forgive me if I correct the
specifics. Short of taking a very close look at the math behind the
possible hand formations it's very difficult to cite what makes one
hold more favorable over another.
In this case, it's the greater potential for 2 Pair and 3 of a Kind on
the JK hold that make it the strongest. All other factor are lesser
in combination.
The unsuited JK hold has an EV of 2.4172 coins on a 5 coin bet. The
unsuited JKA EV is 2.2803. (Note: I've assumed the other cards in
the hand are smaller than 9 so as not to involve interference in
forming straights on the hold.) The contribution of each possible
hand to these EV's is as follows.
JK - HiPr: 1.52 2Pr: 0.44 3K: 0.26 S: 0.14 FH: 0.05 4K: 0.02
JKA- HiPr: 1.61 2Pr: 0.25 3K: 0.12 S: 0.30 FH: 0.00 4K: 0.00
diff HiPr: 0.09 2Pr: 0.19 3K: 0.14 S: (.16) FH: (.05) 4K: (.02)
As indicated, JK actually has a smaller probability of forming a
straight vs JKA. While the JK hold can form either a K-high or A-high
straight vs. just the A-hi of the JKA hold, because there are more
ways for the JK straight to "go wrong", it has a smaller chance of
completion. (I haven't looked at the numbers, but I expect JQ, with 3
possible formations may reverse this relationship.)
So, it's the 2Pr and 3K hold possibilities that are key to making the
JK hold stronger. (Note that HiPr adds a bit more, but not enough to
differentiate the two holds; similarly the reduced EV in the FH and 4K
are insufficient to be a distinction.)
···
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I don't mean to hit anyone over the head with the details -- simply
illustrate that it's often the case that it takes a detailed
examination of the numbers to understand what makes a hold stronger.
These values come from WinPoker,using the "Analyze/Any Hand" feature,
which details the number of possible hand formations from any hold.
To arrive at the numbers I've detailed it's necessary to take these
values to a spreadsheet, divide each number of individual winning hand
possibiliities by the total number of hands that can be formed (to
arrive at a probability; each hold has a different number of total
hand formations, depending upon the number of cards held), and then
multiply by the respective payouts.
One interesting fact from the "Hand Details" summary is that holding
"J", "AK", and "AJ" are all stronger holds than "AJK". "J" is the
weakest of these 3; as you should expect, "AK" and "AJ" are equally
strong.
- Harry