This
strategy was designed to be used for a machine with the
following payoff table (per coin based on maximum coins
bet), although it can be used without much error against
any jacks or better machine. The table also shows the
probability of forming each hand, assuming perfect play,
the contribution to the expected return, and the overall
expected return.
| Jacks
or better |
| Hand |
Payoff |
Probability |
Return |
| Royal
flush |
800 |
0.000025 |
0.019807 |
| Straight
flush |
50 |
0.000109 |
0.005467 |
| Four
of a kind |
25 |
0.002363 |
0.059064 |
| Full
House |
9 |
0.011512 |
0.103610 |
| Flush |
6 |
0.011014 |
0.066087 |
| Straight |
4 |
0.011230 |
0.044919 |
| Three
of a kind |
3 |
0.074449 |
0.223346 |
| Two
pair |
2 |
0.129279 |
0.258558 |
| Jacks
or better |
1 |
0.214585 |
0.214585 |
| Total |
|
0.454566 |
0.995441 |
To use this strategy look up all reasonable ways to
play a hand and choose the play that is highest on the
list. The numbers on the right represent the average
return. These numbers can vary depending on the discards,
those shown are for a typical case. Lets try an example.
If you have a suited ten and jack, an unsuited queen, and
two trash cards, what should you do? Should you keep the
suited ten jack, retaining hope for a flush, straight
flush, and royal flush, or keep the jack and queen,
increasing your odds of forming a high pair. The unsuited
jack and queen appear higher on the list than the suited
ten and jack, thus keep the jack and queen. The numbers in
parenthesis represent the expected return, although the
number listed is just an example and can vary depending on
the discards. The expected returns are not in order
because of the penalty card problem.
From the numbers on the right the expected return of
keeping the 10 and jack is 0.4968 and that of the jack and
queen is 0.4980, thus keeping the jack and queen is the
better play.
While this strategy is 100% accurate (as far as I know)
it is at a cost of being rather long and time consuming to
use. Many players opt to use shorter strategies that only
differ in uncommon and/or borderline plays. I have no
problem with this but personally I like to get every penny
I can out of the machine.
- Pat royal flush (800.0000)
- Pat straight flush (50.0000)
- Pat four of a kind (25.0000)
- 4 to a royal flush (18.4894)
- Pat full house (9.0000)
- Pat flush (6.0000)
- 3 of a kind (4.3025)
- Pat straight (4.0000)
- 4 to a straight flush (3.5319)
- Two pair (2.59574)
- High pair (1.5365)
- 3 to a royal flush (1.4995)
- 4 to a flush (1.2766)
- 4 to an outside straight with 3 high cards (0.8723)
- Low pair (0.8237)
- 4 to an outside straight with 2 high cards (0.8085)
- 4 to an outside straight with 1 high cards (0.7447)
- 3 to a straight flush, spread 3, 1 high cards
(0.7354)
- 4 to an outside straight with 0 high cards (0.6809)
- 3 to a straight flush, spread 5, 2 high cards
(0.6429)
- 3 to a straight flush, spread 4, 1 high card
(0.6392)
- 3 to a straight flush, spread 3, 0 high cards
(0.6207)
- 2 suited high cards, queen highest (0.6004)
- 4 to an inside straight, 4 high cards (0.5957)
- 2 suited high cards, king highest (0.5821)
- 2 suited high cards, ace highest (0.5678)
- 3 to a straight flush, spread 5, 1 high card
(0.5430)
- 4 to an inside straight, 3 high cards (0.5319) A
- 3 to a straight flush, spread 4, 0 high cards
(0.5245)
- 2 unsuited high cards queen highest (0.4980)
- 2 to a royal flush, 10 and jack (0.4968) B
- 2 unsuited high cards king highest (0.4862)
- 2 to a royal flush, 10 and king (0.474869) C
- 2 unsuited high cards ace highest (0.474314)
- 4 to an inside straight, 2 high cards (0.4681)
- 2 to a royal flush, 10 and queen (0.4619)
- Jack only (0.4584)
- 3 unsuited high cards ace highest (0.4561)
- Queen only (0.466224)
- King only (0.463802)
- Ace only (0.465102)
- 2 to a royal flush, 10 and ace (0.460561)
- 3 to a straight flush, spread 5, 0 high cards
(0.4431)
- 4 to an inside straight, 1 high card (0.4043)
- Garbage, discard everything (0.3597)
- 4 to an inside straight, 0 high cards (0.3404)
Rare
Exceptions:
| A |
3
to a straight flush, spread 5, with 1 high card
vs. 4 to an inside straight, with 3 high cards:
Normally the 3 to a straight flush is the better
play however if you must discard a straight
penalty card then go for the straight. For example
if ace,king,queen,10,9 where the king,10,and 9 are
suited. |
| B |
Suited
10 and jack vs. an unsuited jack and king: If
there is no flush penalty card then keeping the 10
and jack then that is the better play, otherwise
keep the jack and king. |
| C |
Suited
10, king vs. king only: Normally the suited ten
and king is better than the king alone, however if
you must discard a 9 and a flush penalty card then
hold the king only. |
Terms:
- Outside straight: An open ended straight that
can be completed at either end, such as (7,8,9,10).
- Inside straight: A straight with a missing
inside card, such as (6,7,9,10).
- Penalty card: Sometimes one must discard a
potentially useful card. For example if you had an
unsuited 10, jack, and queen the ten would be called a
penalty card since you should discard it despite the
fact it could be beneficial if you kept it.
Methodology
To determine the above strategy I created a program can
determine the expected return of the best play of any
hand. The way it works is to consider all 32 ways to play
a hand. For every play the program systematically scores
the held cards with every possible set of discards and
averages the results. The play that yields the greatest
average is determined to be the best play and the specific
statistics for that play are displayed. The program can
also show the statistics for non-optimal plays. Using this
program, it was then a time consuming task to try numerous
borderline hands and rank them in order of expected
return.
About the Author:
Michael Shackleford (a.k.a. The Wizard of Odds) makes his
living as an Actuary for the Social Security
Administration. Visit
his web site here
|
